text:
User talk:Roughcopy

(Red4tribe (talk) 22:13, 22 May 2008 (UTC))

Re: Portfolio Assessments
Joseph, thanks for reading and responding to my article about portolio assessments. I wanted to let you know that the Portfolio vs. Standardized Testing list came from an article listed in the reference section:

Birgin, Osman and Adnan Baki. (2007). The Use of Portfolio to Assess Student's Performance. Journal of Turkish Science Education, vol. 4 (2), 75-90.

I didn't want you to think this was just my opinion. I selected to leave my personal opinions out of this article to remain in line with the neutral point of view. I may move the reference around a bit to make sure it's more visible. The reference appears before the "chart" - I put quotes there because I couldn't figure out how to turn the list into a chart in time for deadline.

Thanks again for your response! Mworr002 (talk) 17:40, 29 July 2008 (UTC)

Peer Reviews
Any One? ;)

Hi Joseph (roughcopy)! Thank you for the very positive "About the Author" response! I'm sorry it's taken me so long to answer your questions, but as you mentioned earlier about me "flirting with insanity"...well, I guess you could say I'm married to it (and NO, I don't mean my husband. He's quite sane. Opposites attract, I guess). Here were your questions, since it's taken me so long to respond, I thought you may have forgotten some of them... I'd like to ask you a few questions, if you don't mind. Why do you want the student to see you as the same person in and out of the classroom? I mean I understand wanting the student to look up to you as a person, but shouldn't there be a destinction between you as the teacher and you as another adult or friend? Would this affect your sense of professionalism? 'I think to be someone you're not, outside of the classroom, would affect my'' sense of professionalism. I guess I put that statement in there b/c I get so tired of teachers acting so prim & proper at school, just to hear them talking who knows what about their latest escapades in the breakroom. It just gives me a deflated feeling to realize they are not who they want others to see them as & I don't want that for myself. I want others to see me as someone who is genuine & someone who tries, but who is not perfect at the same time.'''

How do you feel about the standardized tests. '''I'm not a fond advocate of the SOLs. I'm sure there are other appropriate ways of gauging our students' strengths & abilities. They do not take into consideration that each child is different. Learns different & tests different. Administrators in many schools do make allowances for these differences, though, by setting aside different test locations, etc. to help meet those students' needs. It just seems like we're cramming a set amount of data down our kids' throats & aren't able to give more time to other educational instruments that are just as important (like learning a second language, music, or art experiences). I'm VERY fond of field trips & other ways of teaching kids outside the every day classroom. One of the positives that comes from the SOLs is that they keep teachers accountable, which is very much needed in today's lazy society.'''

I have one final question for you. Since you plan to work in a private school, are there any programs you know of that can help meet your desired technology wishes. I did not know of any but I didn't know if you had looked into it.'''Each school district/county should have a technology "guru" who offers training throughout the year, for the public schools at least. If you're planning on teaching in a public school, I would seek out whomever that person is for your county & stay informed about upcoming training they may offer throughout the year &/or summer. As for a private school setting, I plan to continue taking classes at the university level, to keep me fresh. This would be probably be at my own expense (not that I would have the money to take very many!) or many Christian schools offer training to keep their teachers "fresh", so I would definitely jump on that band wagon. I have based our family expenses on what I am making with my current two jobs. I estimate that I would make a certain amount more once I become a teacher (whether in a private OR public school setting) & plan to set that additional money aside to pay for classes (&/or replace my car). My husband is VERY supportive of most of my decisions in this area. I'd say any fundraisers I would want to do with my classes would go toward field trips or other "fun" learning experiences for the students. Hope this helps clearify a few things! :-) Thanks again for your wonderful comments!'''

Hi Joseph!
What a fascinating background in and with music you have! I think that that kind of passion for music will enable you to become an excellent teacher. When I think back on my educational experiences as a student, and what has helped shape my interests today, I can vividly remember my teachers who truly loved the subjects they were teaching. Also, by "making learning fun", you will certainly inspire more than just the students who are musically talented.

Very insightful userpage!

-Elizabeth Eschm001 (talk) 01:32, 2 June 2008 (UTC)

Author Respone Jmetc001 (talk) 20:43, 15 June 2008 (UTC)
Hey Joseph,

Thanks for the comment. I hope you've had a good summer so far. In reply to your earlier post:

'''Well hello Jessica, again. Did your musical dabbling spill over into teaching or what? why do you want to me a teacher? And by the way, you are an amazing cellist (is that the right word). Any way your teaching philosophy, as I understand is the focus on the student right? I agree with everything that you have said. Especially the emotion part when it comes to music ed. I guess we have one up on the rest... just kidding. I did want to ask, how would you give the individual opporutinities for each student to succeed? What types of opporutinites are we talking about here? And one last thing, I know you at least a bit and it is driving me crazy (for some odd reason) but how would your "decorate" your classroom. and oh what level do you want to teach? Roughcopy (talk) 20:48, 1 June 2008 (UTC) '''

Of course my music and teaching philosophy cross over...I'm going into music education. :-) Honestly, I still don't completely know what I want to do in the future, but, the more I study music and education, the more I love the idea of teaching music.  I have always loved people and loved helping others to the best of my ability.  Thus, teaching seems to follow naturally.

Yes, I want to focus on the individual student. Every person in the world is different, and so, is every student. This means that I must be flexible in my teaching styles in order to meet the needs of each student. It is difficult to say how I would best help each student to succeed, since I haven't met my future students. However, I would work to give the students each the opportunity to shine. One way I might do this through activities that encourage confidence, leadership, and creativity. For example, I might do a review session with the students where each student answers a different question or explains an answer to the entire class. If the student is struggling, I would allow another student to help the first student explain the answer. This way, each student is allowed the opportunity to speak in front of the class and be known. This also teaches the students that there peers are available for support, rather than criticism. The opportunities for individual success may be small scale or large scale, depending the group of individuals being taught.

As for my classroom decorations, I would have fun decorating. I love arts and crafts. I would include classroom management information and subject material information in the decorations because this would help direct the classroom focus to the class subject, most likely music. The decorations would need to be colorful and catchy, without being distracting. Being a RA in the Quad, I decorate the residence hall for my residents. I would imagine the concept of decorating would be the same, only a difference in the decoration audience. Like giving a speech, you must always consider your audience.

I have decided what age level I want to teach. I'd actually prefer to try teaching all levels before making a decision. I would like to start with the younger students and work my way up, the basics through the advanced. The advantage of this would be that I could better learn to teach the fundamentals, basically forcing me to learn to break everything down to it's simplest form so that even the youngest students may understand the concepts. If the students understand the basics with a firm foundation, adding more information and making connections between concepts becomes easier. Better to start simple, like in your classroom management philosophy. :-)

I hope that answers your questions. I'm curious, how much longer before you graduate? Do you plan to go straight to graduate school? Did you have a college in mind where you'd like to teach? Well, take care, and don't work too hard this summer.

Jmetc001 (talk) 20:43, 15 June 2008 (UTC)

title: User talk:Roughcopy


text:
User talk:Dino165

Dinosaurs. Dinosaurs are ancient reptiles that lived between 239-70 million years ago in a long,long,long, time age called the mesozoic. Some dinosaurs were long like diplodocus and argentinosaurus. The largest dino I know is the argentinosaurus he is also the heiviest dinosaur! He is 125 tons and 155 feet long! The smallest dino was microraptor. Microraptor was 19.5 inches long and 1.7 pounds! But a tragic day came 70 million years ago when somthing probably killed off the dinosaurs! It killed off the plants, then the plant-eating dinosaurs, then the meat-eating dinos such as the large tyrannosaur appalachiosaurus! But 70 million years later we can find the same appalachiosaurus in the rock and sand then he will be mounted in a museum. Then the appalachiosaurus will be studyed and stared at for a very,very,very long time.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Dino165


text: Cookbook talk:Chili con Bambi

title: Cookbook talk:Chili con Bambi


text: Cookbook talk:Venison Bolognese Sauce

title: Cookbook talk:Venison Bolognese Sauce


text:
Algorithm Implementation/Miscellaneous/Base64

The traditional (MIME) base64 encoding and decoding processes are fairly simple to implement. Here an example using JavaScript is given, including the MIME/etc required line breaks at particular line lengths. It is worth noting however, that many base64 functions (e.g. in PHP) return base64 encoded strings without the line breaks, as the line breaks can be inserted easily after encoding, and many times the base64 encoding is desired only for safely transferring data via XML or inserting into a database, etc. — times when the line breaks are known to be unnecessary and therefore undesirable. The newline inserting and removing in these functions here can easily be commented out (they are each only one line in the respective functions) if they are not needed.

An array of the base 64 characters is necessary for encoding, such as: And decoding will require the inverse list (swap the indices for the values), such as:

Note that in real implementations, it is better to explicitly list the entire array/hash for each list above — the one-liners here are given to demonstrate the concept as directly as possible, rather than being the ideal in practice.

Javascript
The base64 encoding function:

Java
The base64 encoding function:

An alternate implementation could use Java Streams. For example, one could create a Base64EncoderStream, inheriting from FilterOutputStream.

Ruby
Base64 module documentation show the shortest way to encode a string:

Ruby version, implemented as a tool that encodes the file specified on the command line:

C
C version of the base64 encoding function:

C++
Below is a C++ version of Base64Encode. This code is released into public domain.

Javascript
The base64 decoding function: The above implementation is best with a language like JavaScript that handles string concatenation of arbitrary length strings very efficiently. Other languages (e.g. C) will work much more efficiently by allocating memory for a new string/array of the appropriate size (the output string length is easily calculated from the input string at the very beginning) and then simply setting each character index, as opposed to concatenation.

Java
The base64 decoding function:

An alternate implementation could use Java Streams. For example, one could create a Base64DecoderStream, inheriting from FilterInputStream.

C
Counterpart C decoder for the encoder above. This code is also public domain. This solution has been optimized using pointer math and a look-up table. This algorithm handles multiple encoding formats: with and without line breaks, with and without whitespace, and with and without padding characters.

C++
Counterpart C++ decoder for the encoder above. This code is also public domain. While a more general solution is possible by using a table similar to that in encoding, and then finding the index of the character we want in that table, and while that solution is much simpler, this version with the if-else-if ladder is faster because it does not need to run as many character comparisons to find the right character to use. The alphabet specific parts of this code are pretty much the same across most base64 alphabets however.

Items that need changed to accommodate the base64url encoding are commented as such.


text:
Algorithm Implementation/Miscellaneous

 * /Base64/
 * /N-Queens/
 * Computer Programming/Hello world
 * Transwiki:Create bit mask algorithm

title: Algorithm Implementation/Miscellaneous


text:
User:Dino165

Tyrannosaurus. A large predotory dinosaur around 35.00-39.68 feet long and weighed 6.88 tons. T-rex was a cretaceous dinosaur. It lived 73.99-70.00 million years ago. Tyrannosaurus had 50 7.7 inch teeth. The front 5 teeth in its 4.98 foot mouth were scooping meat,skin, and organs while the rest of its teeth were for crunching bone out of its victim! T-rex ate bugs and lizards while it was young, at teen age it consumed duck-billed dinos and when it was old it barly hunted but when it did it would hunt triceratops and scavenged.

title: User:Dino165


text:
Talk:Chess/Opening theory table

Merge with existing topic
Merge with Chess Opening Theory/Theory table. SunCreator (talk) 11:09, 23 May 2008 (UTC)✅

=/+
what does =/+ mean please? --Arkelweis (talk) 09:57, 7 November 2009 (UTC)
 * It means black is better. SunCreator (talk) 11:57, 10 November 2009 (UTC)
 * and the "∞"? 83.6.155.108 (talk) 00:11, 6 January 2010 (UTC)

What does "∞/=" mean? Could someone add a section or sentence explaining ∞?
 * This is neithera section or sentence, but a reference: Special/Permalink/1019527968
 * I added a wikipedia link that explains the symbols with a little more detail (apologies if this is bad practice here). Basically "∞" means "Unclear", and "=/∞" means "Compensation". RequestSelector (discuss • contribs) 19:16, 19 November 2022 (UTC)
 * Whoops, I just realized there's an existing link to regarding specific positions in that section that contains similar content. I'll leave it as is for now in case it's helpful. Another (perhaps over the top) option would be to transclude the table with . RequestSelector (discuss • contribs) 18:30, 20 November 2022 (UTC)

red moves?
I do not see an explaination for different colors.

ellipsis
why is ... a good idea for a table? Is it just the lazy typist? Seems like a bad choice to me. If the table is very large, the reader has to continually keep moving their eyes up the initial referent if they forget what the referent was. More work for the reader. Ish ishwar (discuss • contribs) 03:26, 26 August 2017 (UTC)

oh, and another issue is that ... has actually two meanings! One is a prefix meaning Black's move. The other is this see-first-mention-in-same-column-inside-of-table variable. Ish ishwar (discuss • contribs) 03:30, 26 August 2017 (UTC)

title: Talk:Chess/Opening theory table


text:
Khepera III Toolbox/Writing your Own Modules

If you add want to use the same piece of code in multiple programs (i.e. a sensor board interface, or a common algorithm), it may be useful to create a module. Such a module can be included and used in all your programs.

The Khepera III Toolbox provides you with a module template and a Makefile.

Preparing the Environment
Make sure that you have a working development environment as described in Writing your Own Programs.

Creating a New Module
To create a new module, type cd my_khepera3_module_directory k3-create-module my_new_module where my_new_module denotes the name of your module. This creates a folder my_new_module with a header and an implementation file (my_new_module.h ,my_new_module.c). A Makefile is created as well.

To check if everything is all right, try to compile the new module: cd my_new_module make This should create an archive my_new_module.a.

You can now edit the files my_new_module.h and my_new_module.c. In case your module is more complex, you can of course create additional my_new_module_*.c and my_new_module_*.h files (note the prefix my_new_module_!). All these additional files will automatically be compiled and available to the programs including the module.

Testing the Module
A module cannot be executed by itself. To test a module, you must create a program that includes this module. To include your module, add the following line to the Makefile of your program: MY_MODULES := path/to/my_new_module This will tell the compiler to look for the *.h files of that module, and to integrate the my_new_module.a archive into the program. (If your program needs multiple modules, simply add them on the same line. Note that all module folders must reside at the same location.)

Now, you can use the functions of the module in your program, and test them therewith. Note that whenever you change code in your module, you need to first recompile the module, and then recompile the program.

Renaming a Module
The name of the binary executable can be changed in the Makefile by modifying the line ARCHIVE := my_new_module.a Note that the archive should always have the same name as the project folder, suffixed by .a.

Including Modules in a Module
A module can include other modules. To include standard modules, add the following line (with the modules you want to add, of course) to the Makefile: MODULES := khepera3 commandline i2cal To include other modules that you wrote on your own, add MY_MODULES := path/to/my_other_module_1 path/to/my_other_module_2 to the Makefile.

SVN / Versioning System Issues
Just as for programs, we strongly suggest using a versioning system such as SVN (Subversion) to store your Khepera III modules. SVN is probably the most prominent system today, and easy to set up and use.

If you are using a versioning system, you should check in the following files of your module:
 * All source files (*.c, *.h)
 * The Makefile
 * The compiled archive (*.a) (such that a compiled copy is available for those who don't have a compiler set up)
 * Any additional files that the program needs

You should not check in:
 * Intermediate compilation files (*.o, *.d)


text: Template talk:Chess Opening Theory/Table


text:
XHTML/History of XHTML

XHTML 1.0
This was the first XHTML standard to be released. It was created to redesign HTML 4.0 to act more like an XML file. To make the change from HTML 4.0 to XHTML 1.0, there was three standards: strict, transitional and frameset. XHTML has the capability to be used with XML files and applications that work with XML files.
 * Strict was the standard that accepted the full XHTML standard.
 * Transitional was used by developers to either make it easier to migrate over from HTML 4.0 or to allow certain standards from HTML version 4.01 Transitional in the XHTML file.
 * Framset was very simply the standard that allowed frames in the XHTML file.

XHTML 1.1
XHTML 1.0 was made with the intention of allowing developers to move from HTML standards to XHTML standards more easily.

XHTML 2.0
XHTML 2.0 was drafted between August 2002 and July 2006. It was designed to supersede HTML 4.0 and XHTML 1.1. In July 2009, World Wide Web Consortium announced plans to discontinue work on XHTML 2.0 by the end of the year in order to focus its effort on HTML 5. HTML 5 is expected to include a XML scheme in place of XHTML 2.0 currently called XHTML 5.

XHTML 5.0
Broadly similar to HTML5.

title: XHTML/History of XHTML


text:
Talk:XHTML/History of XHTML

The content will still need to be added, I will try to get around to it as soon as I can. --61.88.131.149 (talk) 00:25, 23 May 2008 (UTC)

title: Talk:XHTML/History of XHTML


text:
User:Mblount84

Hello All!!

My name is Martin, you TA's know me as Charles, and I am currently teaching band and music theory at Woodside High School in Newport News. Woodside is especially dear to me because I was fortunate enough to go there as a student! Everyone asks me, "Isn't it weird to teach where you once sat in class?". I tell them it's actually preferable - I have never gotten lost in the building :-)

I am working under a provisional license because my major in college was music performance, rather than education. My first love will always be performing (trombone), but I felt it was more important to come back to this area and teach young people how to be good performers.

This leads me to my philosophy on education, which at the moment seems more like a philosophy on living in general. Having already taught for almost 2 years now, I can see the divide that exists between students of different cultures. The main effect of this divide, in my opinion, is that most students are not coming to high school prepared mentally for the amount of study required to be successful. Many highly talented students do just enough to barely "get by" in everything they do! It is my philosophy that the classroom atmosphere must be as organized and calming as possible because that atmosphere can likely be absent at the students' homes.

In this organized environment that is suitable for learning, some creativity must come into play for students to remain engaged. It is through varied instructional practices that this is accomplished. Some musical subjects lend themselves to the lecture format of instruction (music history/appreciation), while units on keyboard and aural skills require a hands on/ears on approach. Many students develop these particular skills only through self-discovery, which can be an instructional model itself if guided by a competent teacher.

Technology implementation in the music classroom is a necessity for 21st century learners. Because of the adolescent addiction to websites like Myspace and YouTube, many students learn more naturally through the use of computers. Music education software is an expensive but important part of the music classroom. In a band class, technology must be utilized to record and playback student performances. Through self-critique and listening to professional recording, mastery of an instrument can increase exponentially. In the theory classroom, ear training software and the keyboard-computer midi links can be used to put the theoretical side of music into sound.

Assessment in the music classroom is quite simple. A student must be able to play the music or write the music required of him or her. I have sometimes found recitation to be another form of assessment useful.

This is first philosophy I have ever been asked to write, and I am extremely curious to know what I would have written had this been a requirement two years before I began teaching!

title: User:Mblount84


text: Wikibooks talk:Chess/Assessment

title: Wikibooks talk:Chess/Assessment


text:
User talk:Aquenvil

Amywilliams (talk) 03:16, 29 May 2008 (UTC)

I really enjoyed reading all about you and your philosophy. You seem to be very passionate about your goal to be a P.E. teacher and that is just what our students need! I know my two favorite things in school were recess and PE, especially when I was in Elementary school. I firmly believe that in order to be a good teacher you must have passion and that will, in turn, fall over onto the students. The will then feel important, loved and energized. I ran track in college and always felt PE should be every day and not just 2 days a week as it is here in No. VA. Not only are you correct about the obesity concern, but the students always come back form PE more chilled out, better able to focus and healthier on top of that! You already know what is important to be a good teacher and that, to me, seems to be half the battle some teachers struggle with every year. I don't have to wish you luck as I feel things will go very well for you when you become a teacher. Your positive attitude will let you soar in hard times.

Amy Williams

I really think you have a great outlook on waht you want to do and we need more people like you in today's schools. Someone like you may get kids a lot more intrested in their own health. You seem to really have a passion for making a difference and i wish the best of luck to you

Joseph Lucente JLuce004 (talk) 18:42, 3 June 2008 (UTC)

Good evening! It's exciting to see a pre-service P.E. teacher with such a love and concern for student wellness. As a current teacher, I have noticed that the subject you hold in such high regard is not appreciated like it should be. Many students do not have a teacher that inspires them or makes them feel that physical education is a course that should be required in school. Inserting technology and literacy into your classroom are sure to make your students, and sometimes just as importantly, your administrators take you seriously. I wish you the best of luck!

--Mblount84 (talk) 03:10, 3 June 2008 (UTC)

title: User talk:Aquenvil


text:
Talk:Introduction to World Issues/Terrorism

5i,ks ,saei8bmx78bn23 kl3782q4wm,b7m3u6mqwu7km234b76376wq3


text:
Template talk:Code:Basic

Possible bug?
This:

a b

yields this:

a b

…which is odd. The style attribute is just an example; it seems to fail with any attributes. — Sam Wilson (Australia) 01:50, 23 May 2008 (UTC)

title: Template talk:Code:Basic


text:
User talk:61.88.131.149

(Red4tribe (talk) 01:53, 23 May 2008 (UTC))

title: User talk:61.88.131.149


text:
HyperText Markup Language/Tag List/caption

Use
Makes a caption for a table. This is depreciated.

More info
A link to a page with more info, then the name of the site the info came from (i.e. W3Schools).


text:
Windows Programming/APIs

 * Telephony Application Programming Interface (TAPI) provides computer telephony integration and enables PCs running Microsoft Windows to use telephone services.

title: Windows Programming/APIs


text:
User talk:70.69.169.235

(Red4tribe (talk) 03:57, 23 May 2008 (UTC))

title: User talk:70.69.169.235


text:
User talk:Htmldash2

(Red4tribe (talk) 03:58, 23 May 2008 (UTC))

title: User talk:Htmldash2


text:
Real Analysis/Landau notation

The Landau notation is an amazing tool applicable in all of real analysis. The reason it is so convenient and widely used is because it underlines a key principle of real analysis, namely estimation. Loosely speaking, the Landau notation introduces two operators which can be called the "order of magnitude" operators, which essentially compare the magnitude of two given functions.

The little-o
The little-o  provides a function that is of lower order of magnitude than a given function, that is the function $$o(g(x))$$ is of a lower order than the function $$g(x)$$. Formally,

Definition
Let $$A\subseteq\mathbb{R}$$ and let $$c\in\mathbb{R}$$

Let $$f,g:A\to\mathbb{R}$$

If $$\lim_{x\to c}\frac{f(x)}{g(x)}=0$$ then we say that

"As $$x\to c$$, $$f(x)=o(g(x))$$"

Examples

 * As $$x\to\infty$$, (and $$m0$$ such that $$\lim_{x\to c}\left| \frac{f(x)}{g(x)}\right| <M$$ then we say that

"As $$x\to c$$, $$f(x)=O(g(x))$$"

Examples

 * As $$x\to 0$$, $$\sin x=O(x)$$
 * As $$x\to \tfrac{\pi}{2}$$, $$\sin x=O(1)$$

Applications
We will now consider few examples which demonstrate the power of this notation.

Differentiability
Let $$f: \mathcal{U} \subseteq \mathbb{R} \to\mathbb{R}$$ and $$ x_0 \in \mathcal{U}$$.

Then $$f$$ is differentiable at $$x_0$$ if and only if

There exists a $$\lambda \in\mathbb{R}$$ such that as $$x\to x_0$$, $$f(x) = f(x_0) + \lambda(x-x_0)+o\left( |x-x_0|\right)$$.

Mean Value Theorem
Let $$f:[a,x]\to\mathbb{R}$$ be differentiable on $$[a,b]$$. Then,

As $$x\to a$$, $$f(x)=f(a)+O(x-a)$$

Taylor's Theorem
Let $$f:[a,x]\to\mathbb{R}$$ be n-times differentiable on $$[a,b]$$. Then,

As $$x\to a$$, $$f(x)=f(a)+\tfrac{(x-a)f'(a)}{1!}+\tfrac{(x-a)^2f''(a)}{2!}+\ldots+\tfrac{(x-a)^{n-1}f^{(n-1)}(a)}{(n-1)!}+O\left( (x-a)^n\right)$$

title: Real Analysis/Landau notation


text:
Woodworking/Sharpening

The ability to sharpen an edge is of utmost importance in woodwork. Without it, the woodworker is destined to hours of frustration, hard work, and ultimately a poor result; luckily, there are many techniques for sharpening that are easy to learn, require no expensive machinery, and with practice will produce exceptionally sharp edges. The key here, however, is practice: you can not learn to sharpen an edge properly by reading a book or even being shown how to do it. These will help, and are even necessary to begin, but it is relentless repetition—many hours spent at the bench, improving at an almost imperceptible pace—that will ensure that you can sharpen tools properly. Thankfully, this is 'all' it will take!

When we say that expensive machinery is unnecessary, we do not mean to imply that a good-quality grinder (for shaping chisels, plane blades, etc.), or specialised honing system, would not be a valuable addition to the workshop. It is however quite possible to produce the same (or better) results with lower-cost tools that are much more convenient for the small workshop, and easier to obtain for the amateur woodworker.

To sharpen all of the standard, straight-edged tools (and saws are included in this category, because their actual cutting edges are straight), one needs a surprisingly small number of tools and materials. These are as follows, and their purchase, preparation, use, and storage will be elaborated upon below.

 * Saws:
 * files, flat and triangular, of assorted sizes and coarseness;
 * saw set;
 * …
 * Plane blades, chisels, etc.:
 * three medium-density water stones;
 * a higher-density final honing stone;
 * a range of small triangular angle gauges (made of plastic, wood, or something else non-corrosive);
 * …
 * …

Blade tools
Sharpening of all edge tools is much the same: shape the edge (with a file or grinder); hone it (with an oil- or water-stone); then strop it (with a leather strop, MDF polishing board, or even the flabby bit of the hand). With good steel and flat stones, this process will yield the sharpest of edges every time—sharp enough to shave the hairs off your head with feather-like pressure.

Flattening Stones
All stones, be they waterstones or oilstones, are required to be of the utmost flatness before they can be used for sharpening. Thankfully, there is a simple and utterly reliable method for accomplishing this.

Procure three stones of equal or similar grit and hardness; label these 'A', 'B', and 'C' with a permanent marker at one end of each of their long edges. Also mark the back sides of each stone with an 'X' in the middle (we only need to use one face). These marks will make it possible to identify each stone and also orient them end-to-end in a repeatable fashion.

The flattening process involves wearing every stone against every other stone in a cycle that ensures that, gradually as it's repeated many times, the high and low areas of each stone are "reduced to grade". This grade is a truely flat surface, and can form the reference against which all tool edges can be measured. (Note that this process does not work with two stones only, and nor does it need more than three.)

To begin, take stones A and B in your left and right hands respectively and with the lettered-edges uppermost. Rub the front faces of these stones together. Quickly, the high points of each stone will be visible as the areas that are being worn away first. Don't try to extend these areas too far yet (this is a process that should be gradual), but instead start to even out the inevitable unevenness of your hand pressure by rotating stone 'B' so that it's label is now facing down and at the closer end of the stone. Rub again in this position. Perhaps 10 forward-and-backward motions in each of these two positions are all that are required; perhaps even fewer, depending on the hardness of your stones.

Next, put down stone 'A', swap stone 'B' to your left hand, and take up stone 'C' with your right. 'B' and 'C' should now be as 'A' and 'B' were when you began, and what you did with them you must now exactly do with this new pair.

There's a pattern here, of course: after wearing B and C together, you move on to C and A — and this is the last step in the cycle before you begin again. Note also that this third time, A is in your right hand (whereas before it was in your left) and this is also part of ensuring that your muscular inequalities do not have inequal effect on the stones.

Carry on in this fashion — A-B, B-C, C-A, A-B, etc. — and slowly the stones will come to match each other's surfaces and in doing so will be perfectly flat.

Saws
This section is concerned with traditional English-style handsaws that cut on the push stroke (as opposed to, for example, Japanese saws).

All work on saws should be done with the saw held firmly in a saw vice, to stop blade flex and chatter; these are annoying and shorten the life of the files. There are many different types of vices, from commercial cast-iron cam-operated models, to the simplest method of two pieces of timber held either side of the blade. Clamp the saw blade in the vice with about 3 or 4mm of blade showing (below the gullets of the teeth).

1. Jointing
Jointing is the process of filing all of a saw's teeth down to a common line, and needs to be done only when the teeth are uneven or incorrectly shaped.

2. Shaping
Correcting the profile of the teeth.

3. Setting
Bending the teeth out to each side of the blade about ¼ the thickness of the blade.

4. Filing
Actually putting the cutting edge to the teeth, working from alternating sides and filing towards the handle.

title: Woodworking/Sharpening


text:
User:Willscrlt/ForMoreSeeWP

 |.

Syntax

 * 1) title: the article title at the other wiki to which you wish to link; defaults to the current SUBPAGENAME (that would be "" for this page)
 * 2) type: defaults to "this topic", but could be "this recipe", "these photos", or "Magicians"
 * 3) site: defaults to "The English Wikipedia", but could be "Wikimedia Commons", "Die deutschsprachige Wikipedia", "Wikispecies", etc.
 * 4) wiki: defaults to "w" (as in "w:Magician (fantasy)"), but could be "commons" (for "commons:Category:Magicians") or "m:de" (for "m:de:Zauberkünstler")
 * 5) what: defaults to "has more information about", but could be "includes a biography on", "has photographs of", etc.

Related templates
User:Willscrlt/ForCocktailSeeWP

title: User:Willscrlt/ForMoreSeeWP


text:
User:Willscrlt/ForCocktailSeeWP

Syntax

 * 1) cocktail: the article title at the other wiki for the drink to which you wish to link; defaults to the current SUBPAGENAME (that would be "" for this page)
 * 2) name: defaults to the cocktail if given; this is the "pretty name" of the article that humans read

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User:Willscrlt/ForMoreSeeWP

title: User:Willscrlt/ForCocktailSeeWP


text:
User:Jkaduda

Areas of Expertise ·	ICT Project Management	·	ICT Human Resources Staffing, Mentoring, Training ·	Data Centre Planning, Building, Managing	·	ICT Regulatory Business Functions Management ·	TCP/IP Network Management (Netware/Windows)	·	VSAT Assessment Installation requirements

ICT Management Skills Enterprise, ICT Budgeting monitoring Data Centre Set-up and Management ICT Training, Infrastructure Consolidation, Service Level Management Managing Telecommunication resources

Professional Experience ICT Infrastructure Manager KEMRI/Wellcome Trust Research Programme

·	ICT Infrastructure Manager

Published Articles ·	A Digital Library In A Rural Malaria Research Centre In Africa; The Kenyan Experience (IILR) (http://www.asis.org/SIG/SIGIII/paper_contest/2003/2003winners.htm )

title: User:Jkaduda


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User:Jackmcgarry

Jack McGarry Nashua, NH

Biography: Born: Arlington, MA - Sept 11, 1955 Military: USMC (Marine Security Guard Bn) USARNG (972nd Military Police Co.) Employment: Current: Vela Systems, Burlington, MA (SQA Manager) Contact: email: jackmcgarry@yahoo.com

title: User:Jackmcgarry


text:
User talk:79.216.80.12

Red4tribe (talk) 12:47, 23 May 2008 (UTC)

title: User talk:79.216.80.12


text:
User talk:Allispaul

PAUL! --82.1.253.14 (talk) 13:16, 23 May 2008 (UTC)

title: User talk:Allispaul


text:
User:Evonx002

I am an undergraduate college student who is majoring in math education. My plan is to become a teacher (hopefully without having to go another semester on top of the four years). My mother is also a teacher and I her work has inspired me to do the same.

I have been doing my student observation hours, which has helped form my philosophy of education. I want my students to understand the material and the applications of it. Activities are a good way of keeping a student's attention. I believe that is the teacher's job to make the students learn the material. I think student's should be grouped by ability. That way those that need extra help can get it, while those who understand it will not have to wait for the slower learners. I think my instruction will have both lecture and group problem solving. I think groups are important, because students can learn through sharing their knowledge. I want my stdents to feel free to ask questions. I think that the classroom should be a relaxed environment. Parents should have up to date progress reports. If the student is not motivated, the parents should be able to see that and do something about it. My students should learn how to use a TI-83 calculator, which will make their work easier. Also, m students should be familiar with some math programs on the computer. At the beginning of the year, I plan on giving them a quiz to see what information they remember from their previous math courses. This will give me an idea on what material we should spend more time on.

title: User:Evonx002


text:
User talk:63.88.38.96

Red4tribe (talk) 16:05, 23 May 2008 (UTC)

title: User talk:63.88.38.96


text:
User talk:69.109.57.84

(Red4tribe (talk) 17:00, 23 May 2008 (UTC))

title: User talk:69.109.57.84


text:
User talk:169.227.254.3

(Red4tribe (talk) 18:18, 23 May 2008 (UTC)

title: User talk:169.227.254.3


text:
Talk:Turing

i have got a problem with Qbasic. I have created a qbasical hangman but when i finish the commands and start with the sub the whole program goes crazy and then stops working.and without the sub it works perfectly. but i need more words so i need sub what am i gonna do??????What can i do with this code. my teacher says do without a sub command but i don't hav any idea without sub what do i do????????????

title: Talk:Turing


text:
User talk:Fabifab

(Red4tribe (talk) 18:55, 23 May 2008 (UTC))

title: User talk:Fabifab


text:
User talk:Lauren 254

Welcome to Wikibooks, Lauren 254!

Butterfly Meadow Ranch
Wikibooks is for textbooks, not fiction. If you would like to write fiction, there is fiction. – Mike.lifeguard  | talk 22:20, 23 May 2008 (UTC)

title: User talk:Lauren 254


text:
Talk:Muggles' Guide to Harry Potter/Books/Prisoner of Azkaban/Chapter 2

Answer to Question # 3. Harry didn't get his uncle to sign the permission slip. The book never says he does and, later on, he tries to get Professor McGonagall to sign it, but she won't. Then, at the end of the book, Professor Dumbledore says that Sirius gives him permission to go.
 * Thank you for not answering the study questions in the book proper. Yes, that is mostly correct (Sirius provides his own permission letter, Dumbledore has nothing to do with it); but this being a textbook for students, we want to leave the questions for the students to work on. Chazz (talk) 22:43, 12 November 2008 (UTC)


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User:BronzeDragonet

Figured I'd be nice to anyone who decided to stop by my Profile and say Hi to you first.

Heres a link to my wikipedia account BronzeDragonet

Currently Learning

 * http://en.wikibooks.org/wiki/C_Programming

Future Possibilities
http://en.wikibooks.org/wiki/Wikibooks:Computing_department

 * http://en.wikibooks.org/wiki/JavaScript
 * http://en.wikibooks.org/wiki/XHTML
 * http://en.wikibooks.org/wiki/PHP_Programming

http://en.wikibooks.org/wiki/Wikibooks:Languages_bookshelf

 * http://en.wikibooks.org/wiki/Latin
 * http://en.wikibooks.org/wiki/Irish

Can't Learn Due to Construction/Unfinished book
http://en.wikibooks.org/wiki/Japanese

http://en.wikibooks.org/wiki/Programming:HTML

http://en.wikibooks.org/wiki/CSS_Programming

I'll try to fix things up if I see errors or graffiti/vandalism or if things need to be finalized into table format to become suitable for the public.

title: User:BronzeDragonet


text:
User talk:88.90.225.93

(Red4tribe (talk) 22:05, 23 May 2008 (UTC))

title: User talk:88.90.225.93


text: User talk:216.168.123.22

title: User talk:216.168.123.22


text:
User talk:Slacker876

Welcome to Wikibooks, Slacker876!

Test edits
If you want to make test edits, please do so in the sandbox. – Mike.lifeguard  | talk 22:23, 23 May 2008 (UTC)

title: User talk:Slacker876


text:
File talk:Types.png

Replaced on C++ Programming
into

To address issue "Fix the image showing a long has bits 0 to 32, and not 0 to 31" reported on this edit by anonymous user. --Panic (talk) 22:57, 23 May 2008 (UTC)

title: File talk:Types.png


text: User talk:Kika321

title: User talk:Kika321


text:
Chess Opening Theory/1. e4/1...c5/2. e5

= Sicilian with 2.e5 = This rarely seen move has some logical basis. The pawn on e5 prevents Black playing Nf6 and so the Black Kingside is cramped for a while. White can support the e5 pawn with Nf3. The drawback for White of 2.e5 is no added pressure is brought to the center, especially on d4 or d5 and the extra tempo allows Black various options, the main one is play Nc6 which hinders White's d4 move as well as attacking the as yet undefended e5 pawn.

Theory table
.

1.e4 c5 2.e5

title: Chess Opening Theory/1. e4/1...c5/2. e5


text:
Talk:Chess Opening Theory/1. e4/1...c5/2. e5

It's interesting that this is here. The treatment of this variation in the article is misconceived. This is a gambit. White is going to give up the e5 pawn and have a d4 pawn instead, and that consumes 3 moves for Black. 1. e4 c5 2. e5 Nc6 3. c3 Nxe5 4. d4. After the Black N moves, White must play Nc3 or he will be lost. However, after that, White has free development and can sometimes prevent Black from castling and Black's pieces will not be well placed. Black may think he's attacking, but his moves may turn out to be losers as White seizes the initiative. It is, of course, unsound, like most gambits. It will work against good players who have never seen it before, if you know the ideas. 72.182.33.219 (discuss) 02:06, 29 March 2015 (UTC) Eric


text:
User talk:Obermeic

Red4tribe (talk) 23:23, 23 May 2008 (UTC)

title: User talk:Obermeic


text:
User talk:62.65.193.62

Red4tribe (talk) 23:50, 23 May 2008 (UTC)

title: User talk:62.65.193.62


text:
User talk:Mblount84

Red4tribe (talk) 23:51, 23 May 2008 (UTC)

Hello Martin/Charles,

It is great to have an in-service teacher in the class. I know we will all benefit from your experience. I am curious about the role music class can play in illuminating students' different intelligences or talents. In most classes, linguistic and mathematical and of course interpersonal intelligence/talents are rewarded and recognized. In my experience, I have valued courses that are considered vocational such as art, music or even drama for the opportunity they provide for "other" kids to "shine". By "other" I mean the students who often do not get recognition in regular classes because they may not be the best reader or math student or because they are shy. Music, art and other such classes give these kids a chance to showcase their talents and earn a little admiration from their peers. I have never asked a music or art teacher about this phenomenon and I am curious to know if you see this also. Or, perhaps, you are accustomed to seeing these music stars and would be surprised by the scholars of the regular classroom. What has been your experience? Jenniferjkidd (talk) 18:15, 25 May 2008 (UTC)

Hi Martin,

I have also observed highly talented students doing enough to just get by. I am also on provisional license teaching computer networking at the Aviation Academy. Interestingly enough, my first job with NNPS was as a long term substitute teach guitar at DHS. (I am a classic guitarist and bassist). I have discovered this year (my 2nd at AA) that in order to get students performing at or near peak, I have had to spend a lot of time getting to know them better by talking one on one. On one occasion I have had to conference with a student and their parent together. I am now getting better results. Sometimes it works and sometimes it does not work as well as I would like. Mmann019 (talk) 01:20, 2 June 2008 (UTC)

title: User talk:Mblount84


text:
User talk:63.76.209.82

Red4tribe (talk) 23:52, 23 May 2008 (UTC)

title: User talk:63.76.209.82


text:
User talk:Wei-tschie

Red4tribe (talk) 23:53, 23 May 2008 (UTC)

title: User talk:Wei-tschie


text:
User talk:68.110.230.28

Red4tribe (talk) 00:16, 24 May 2008 (UTC)

title: User talk:68.110.230.28


text:
C++ Programming/Code/Standard C Library/Functions/difftime

difftime
The function difftime returns time2 - time1, in seconds.

 * Related topics: asctime - gmtime - localtime - time


text:
User talk:70.160.99.107

Red4tribe (talk) 00:59, 24 May 2008 (UTC)

title: User talk:70.160.99.107


text:
User talk:BronzeDragonet

Welcome, Cysaki!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Red4tribe (talk) 01:03, 24 May 2008 (UTC)

Japanese Wikibook discussion
Hi Cysaki,

I just added a comment on Talk:Japanese which I wanted to draw your attention to as you've been involved in this book. --Swift (talk) 14:40, 10 June 2008 (UTC)

title: User talk:BronzeDragonet


text:
User talk:59.101.33.56

Hello and welcome to Wikibooks. If you would like an article imported here from Wikipedia, please list it on our request for imports page. If you just copy the page, it violates the license, though both are the same, the history must be preserved. Thank you for your cooperation. Mattb112885 (talk to me) 01:40, 24 May 2008 (UTC)

title: User talk:59.101.33.56


text:
User talk:Katdai

Welcome, Katdai! {| style="background:white; border:1px solid #abd5f5;; padding:0px; border-spacing:0px; color: #000000;" ! style="background:#d0e5f5; color: #000000;" | My name is Kat Dai.

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Red4tribe (talk) 01:44, 24 May 2008 (UTC)

title: User talk:Katdai


text:
User talk:Gregory.dulin

Welcome, Gregory.dulin!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally. Red4tribe (talk) 01:47, 24 May 2008 (UTC)

title: User talk:Gregory.dulin


text:
Functional Analysis/Hilbert spaces

A normed space is called a pre-Hilbert space if for each pair $$(x, y)$$ of elements in the space there is a unique complex (or real) number called an inner product of $$x$$ and $$y$$, denoted by $$\langle x, y \rangle$$, subject to the following conditions:
 * (i) The functional $$f(x) = \langle x, y \rangle$$ is linear.
 * (ii) $$\langle x, y \rangle = \overline {\langle y, x \rangle}$$
 * (iii) $$\langle x, x \rangle > 0$$ for every nonzero $$x$$

The inner product in its second variable is not linear but antilinear: i.e., if $$g(y) = \langle x, y \rangle$$, then $$g(\alpha y) = \bar{\alpha} y$$ for scalars $$ \alpha$$. We define $$\| x \| = \langle x, x \rangle^{1/2}$$ and this becomes a norm. Indeed, it is clear that $$\| \alpha x \| = | \alpha | \| x \|$$ and (iii) is the reason that $$\| x \| = 0$$ implies that $$x = 0$$. Finally, the triangular inequality follows from the next lemma.

3.1 Lemma (Schwarz's inequality) $$|\langle x, y \rangle| \le \|x\|\|y\|$$ where the equality holds if and only if we can write $$x = \lambda y$$ for some scalar $$\lambda$$.

If we assume the lemma for a moment, it follows:

since $$\operatorname{Re}(\alpha) \le | \alpha |$$ for any complex number $$\alpha$$
 * $$\| x + y \|^2$$
 * $$= \| x \|^2 + 2 \operatorname{Re} \langle x, y \rangle + \| y \|^2 \le \| x \|^2 + 2 | \langle x, y \rangle | + \| y \|^2 $$
 * $$\le (\| x \| + \| y \|)^2$$
 * }
 * $$\le (\| x \| + \| y \|)^2$$
 * }
 * }

Proof of Lemma: First suppose $$\|x\| = 1$$. If $$\alpha = \overline {\langle x, y \rangle}$$, it then follows:
 * $$0 \le \| \alpha x - y\|^2 = | \alpha |^2 - 2 \operatorname{Re} (\alpha \langle x, y \rangle) + \|y\|^2 = -| \alpha |^2 + \|y\|^2$$

where the equation becomes $$0$$ if and only if $$x = \lambda y$$. Since we may suppose that $$x \ne 0$$, the general case follows easily. $$\square$$

3.2 Theorem A normed linear space is a pre-Hilbert space if and only if $$\|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2 - \|x+y\|^2$$.

Proof: The direct part is clear. To show the converse, we define
 * $$\langle x, y \rangle = 4^{-1} (\|x+y\|^2 - \|x-y\|^2 + i\|x+iy\|^2 - i\|x-iy\|^2)$$.

It is then immediate that $$\langle x, y \rangle = \overline{\langle y, x \rangle}$$, $$\langle -x, y \rangle = -\langle x, y \rangle$$ and $$\langle ix, y \rangle = i \langle x, y \rangle$$. Moreover, since the calculation:

we have: $$\langle x_1 + x_2, y \rangle = \langle x_1, y \rangle + \langle x_2, y \rangle$$. If $$\alpha$$ is a real scalar and $$\alpha_j$$ is a sequence of rational numbers converging to $$\alpha$$, then by continuity and the above, we get: $$\langle \alpha x, y \rangle = \lim_{j \to \infty} \langle \alpha_j x, y \rangle = \alpha \langle x, y \rangle. \square$$
 * $$\|x_1 + x_2 + y\|^2 - \|x_1 + x_2 - y\|^2$$
 * $$=2\|x_1+y\|^2 - 2\|x_1-y\|^2 - \|x_1 - x_2 + y\|^2 - \|x_1 - x_2 - y\|^2$$
 * $$=\sum_{j=1}^2 \|x_j+y\|^2 - \|x_j-y\|^2$$,
 * }
 * $$=\sum_{j=1}^2 \|x_j+y\|^2 - \|x_j-y\|^2$$,
 * }
 * }

3.3 Lemma ''Let $$\mathfrak{H}$$ be a pre-Hilbert. Then $$x_j \to x$$ in norm if and only if for any $$y \in \mathfrak{H}$$ $$\|x_j\| \to \|x\|$$ and $$\langle x_j - x, y \rangle \to 0$$ as $$j \to \infty$$.

Proof: The direct part holds since:
 * $$| \|x_j\| - \|x\| | + |\langle x_j - x, y \rangle| \le \|x_j - x\|(1 + \|y\|) \to 0$$ as $$j \to \infty$$.

Conversely, we have:
 * $$\| x_j - x \|^2 = \|x_j\|^2 - 2\operatorname{Re} \langle x_j, x \rangle + \|x\|^2 \to 0$$ as $$j \to \infty$$

$$\square$$

3.4 Lemma ''Let $$D$$ be a non-empty convex closed subset of a Hilbert space. Then $$D$$ admits a unique element $$z$$ such that
 * $$\| z \| = \inf \{ \|x\|; x \in D \}$$.''

Proof: By $$\delta$$ denote the right-hand side. Since $$D$$ is nonempty, $$\delta > 0$$. For each $$n = 1, 2, ...$$, there is some $$x_n \in D$$ such that $$0 \le \|x_n\| - \delta \le n^{-1}$$. That is, $$\delta = \lim_{n \to \infty} \| x_n \|$$. Since $$D$$ is convex,
 * $${ x_n + x_m \over 2 } \in D$$ and so $$\delta \le {1 \over 2} \|x_n + x_m\|$$.

It follows: This is to say, $$x_n$$ is Cauchy. Since $$D$$ is a closed subset of a complete metric space, whence it is complete, there is a limit $$z \in D$$ with $$\|z\| = \delta$$. The uniqueness follows since if $$\|w\| = \delta$$ we have
 * $$\|z - w\|^2 = 2\|z\|^2 + 2\|w\|^2 - \|z+w\|^2$$

where the right side is $$\le 0$$ for the same reason as before. $$\square$$

The lemma may hold for a certain Banach space that is not a Hilbert space; this question will be investigated in the next chapter.

For a nonempty subset $$E \subset \mathfrak{H}$$, define $$E^\bot$$ to be the intersection of the kernel of the linear functional $$u \mapsto \langle u, v \rangle$$ taken all over $$v \in E$$. (In other words, $$E^\bot$$ is the set of all $$x \in \mathfrak{H}$$ that is orthogonal to every $$y \in E$$.) Since the kernel of a continuous function is closed and the intersection of linear spaces is again a linear space, $$E^\bot$$ is a closed (linear) subspace of $$\mathfrak{H}$$. Finally, if $$x \in E \cap E^\bot$$, then $$0 = \langle x, x \rangle = \|x\|$$ and $$x = 0$$.

3.5 Lemma ''Let $$\mathcal{M}$$ be a linear subspace of a pre-Hilbert space. Then $$z \in \mathcal{M}^\bot$$ if and only if $$\| z \| = \inf \{ \| z + w \| ; w \in \mathcal{M}\}$$.

Proof: (<=). Let $$w \in \mathcal M$$. By our condition, we have that $$\lVert z \rVert \leq \lVert z + w\rVert$$. Squaring both sides gives $$\lVert z \rVert^2 \leq \lVert z + w\rVert^2$$. Expanding this using inner products and rearranging gives $$2\Re \langle z, w \rangle \geq -\lVert w \rVert^2$$. The same thing is true (by the same argument) for $$-w$$, so we get $$2\Re \langle z, -w \rangle \geq -\lVert w \rVert^2$$. This altogether implies that $$-\lVert w \rVert^2 \leq 2\Re \langle z, -w \rangle = -2\Re \langle z, w \rangle \leq \lVert w \rVert^2$$, from which we get $$2|\Re \langle z, w \rangle| \leq \lVert w \rVert^2$$. Consider a real $$\lambda > 0$$; by the same argument we have that $$2|\Re\langle z, w \rangle| \leq \lambda \lVert w\rVert^2$$. Since this is true for all $$\lambda > 0$$, we get $$\Re \langle z, w \rangle = 0$$. Since furthermore we have $$i w \in \mathcal M$$, we have that $$0 = \Re \langle z, iw \rangle = -\Im\langle z, w \rangle$$. We conclude that $$\langle z, w \rangle = 0$$.

(=>) Let $$w \in \mathcal M$$. We have that $$\lVert z + w \rVert^2 = \lVert z \rVert^2 + 2\Re\langle z, w \rangle + \lVert w \rVert^2 = \lVert z \rVert^2 + \lVert w \rVert^2 \geq \lVert z \rVert^2$$. Taking the first and last term in this quality, and applying the square root, gives $$\lVert z + w \rVert \geq \lVert z \rVert$$. Finally, notice that for $$w = 0 \ni \mathcal M$$, the infimum is obtained because $$\lVert z + w \rVert = \lVert z \rVert$$. $$\square$$

3.6 Theorem (orthogonal decomposition) ''Let $$\mathfrak{H}$$ be a Hilbert space and $$\mathcal{M} \subset \mathfrak{H}$$ be a closed subspace. For every $$x \in \mathfrak{H}$$ we can write
 * $$x = y + z$$

where $$y \in \mathcal{M}$$ and $$z \in \mathcal{M}^\bot$$, and $$y$$ and $$z$$ are uniquely determined by $$x$$.''

Proof: Clearly $$x - \mathcal{M}$$ is convex, and it is also closed since a translation of closed set is again closed. Lemma 3.4 now gives a unique element $$y \in \mathcal{M}$$ such that $$\|x - y \| = \inf \{ \|x - w\|; w \in \mathcal{M}\}$$. Let $$z = x - y$$. By Lemma 3.5, $$z \in \mathcal{M}^\bot$$. For the uniqueness, suppose we have written:
 * $$x = y' + z'$$

where $$y' \in \mathcal{M}$$ and $$z' \in \mathcal{M}^\bot$$. By Lemma 3.5, $$\|x - y' \| = \inf \{ \|x - w\|; w \in \mathcal{M}\}$$. But, as noted early, such $$y'$$ must be unique; i.e., $$y' = y$$. $$\square$$

3.7 Corollary ''Let $$\mathcal{M}$$ be a subspace of a Hilbert space $$\mathfrak{H}$$. Then Proof: By continuity, $$\langle x, \overline{\mathcal{M}} \rangle \subset \overline{\langle x, \mathcal{M} \rangle}$$. (Here, $$\langle x, E \rangle$$ denotes the image of the set $$E$$ under the map $$y \mapsto \langle x, y \rangle$$.) This gives:
 * (i) $$\mathcal{M}^\bot = \{0\}$$ if and only if $$\mathcal{M}$$ is dense in $$\mathfrak{H}$$.
 * (ii) $$\mathcal{M}^{\bot\bot} = \overline{\mathcal{M}}$$.
 * $$\mathcal{M}^\bot = \overline{\mathcal{M}}^\bot$$ and so $$\mathfrak{H} = \overline{\mathcal{M}} \oplus \mathcal{M}^\bot$$

by the orthogonal decomposition. (i) follows. Similarly, we have:
 * $$\mathfrak{H} = \mathcal{M}^\bot \oplus \mathcal{M}^{\bot\bot} = \mathcal{M}^\bot \oplus \overline{\mathcal{M}}$$.

Hence, (ii). $$\square$$

3.8 Theorem (representation theorem) ''Every continuous linear functional $$f$$ on a Hilbert space $$\mathfrak{H}$$ has the form:
 * $$f(x) = \langle x, y \rangle$$ with a unique $$y \in \mathcal{M}$$ and $$\|f\| = \|y\|_\mathfrak{H}$$

Proof: Let $$\mathcal{M} = f^{-1} ( \{0\} )$$. Since $$f$$ is continuous, $$\mathcal{M}$$ is closed. If $$\mathcal{M} = \mathfrak{H}$$, then take $$y = 0$$. If not, by Corollary 3.6, there is a nonzero $$z \in \mathfrak{H}$$ orthogonal to $$\mathcal{M}$$. By replacing $$z$$ with $$z \|z\|^{-1}$$ we may suppose that $$\|z\| = 1$$. For any $$x \in \mathfrak{H}$$, since $$zf(x) - f(z)x$$ is in the kernel of $$f$$ and thus is orthogonal to $$z$$, we have:
 * $$0 = \langle z f(x) - f(z) x, z \rangle = \langle z, z \rangle f(x) - \langle f(z)x, z \rangle$$

and so:
 * $$f(x) = \langle x, \overline{f(z)} z \rangle$$

The uniqueness follows since $$\langle x, y_1 \rangle = \langle x, y_2 \rangle$$ for all $$x \in \mathfrak{H}$$ means that $$y_1 - y_2 \in \mathfrak{H}^\bot = \{0\}$$. Finally, we have the identity:
 * $$\|y\| = |\langle {y \over \|y\|}, y \rangle | \le \|f\| \le \|y\|$$

where the last inequality is Schwarz's inequality. $$\square$$

3.9 Exercise Using Lemma 1.6 give an alternative proof of the preceding theorem.

In view of Theorem 3.5, for each $$x \in \mathfrak{H}$$, we can write: $$x = y + z$$ where $$y \in \mathcal{M}$$, a closed subspace of $$\mathfrak{H}$$, and $$z \in \mathcal{M}^\bot$$. Denote each $$y$$, which is uniquely determined by $$x$$, by $$\pi(x)$$. The function $$\pi$$ then turns out to be a linear operator. Indeed, for given $$x_1, x_2 \in \mathfrak{H}$$, we write:
 * $$x_1 = y_1 + z_1, x_2 = y_2 + z_2$$ and $$x_1 + x_2 = y_3 + z_3$$

where $$y_j \in \mathcal{M}$$ and $$z_j \in \mathcal{M}^\bot$$ for $$j = 1, 2, 3$$. By the uniqueness of decomposition
 * $$\pi(x_1) + \pi(x_2) = y_1 + y_2 = y_3 = \pi(x_1 + x_2)$$.

The similar reasoning shows that $$\pi$$ commutes with scalars. Now, for $$x = y + z \in \mathfrak{H}$$ (where $$y \in \mathcal{M}$$ and $$z \in \mathcal{M}^\bot$$), we have:
 * $$\|x\|^2 = \|\pi(x)\|^2 + \|z\|^2 \ge \|\pi(x)\|^2$$

That is, $$\pi$$ is continuous with $$\|\pi\| \le 1$$. In particular, when $$\mathcal{M}$$ is a nonzero space, there is $$x_0 \in \mathcal{M}$$ with $$\pi(x_0) = x_0$$ and $$\|x_0\| = 1$$ and consequently $$\|\pi\| = 1$$. Such $$\pi$$ is called an orthogonal projection (onto $$\mathcal{M}$$).

The next theorem gives an alternative proof of the Hahn-Banach theorem.

3 Theorem ''Let $$\mathcal{M}$$ be a linear (not necessarily closed) subspace of a Hilbert space. Every continuous linear functional on $$\mathcal{M}$$ can be extended to a unique continuous linear functional on $$\mathfrak{H}$$ that has the same norm and vanishes on $$\mathcal{M}^\bot$$.''

Proof: Since $$\mathcal{M}$$ is a dense subset of a Banach space $$\overline{\mathcal{M}}$$, by Theorem 2.something, we can uniquely extend $$f$$ so that it is continuous on $$\overline{\mathcal{M}}$$. Define $$g = f \circ \pi_{\overline{\mathcal{M}}}$$. By the same argument used in the proof of Theorem 2.something (Hahn-Banach) and the fact that $$\|\pi_\mathcal{F}\|= 1$$, we obtain $$\|f\| = \|g\|$$. Since $$g = 0$$ on $$\mathcal{M}^\bot$$, it remains to show the uniqueness. For this, let $$h$$ be another extension with the desired properties. Since the kernel of $$f - h$$ is closed and thus contain $$\overline{\mathcal{M}}$$, $$f = h$$ on $$\overline{\mathcal{M}}$$. Hence, for any $$x \in \mathfrak{H}$$,
 * $$h(x) = (h \circ \pi_{\overline{\mathcal{M}}})x = (f \circ \pi_{\overline{\mathcal{M}}})x = g(x)$$.

The extension $$g$$ is thus unique. $$\square$$

3 Theorem ''Let $$\mathcal{M}_n$$ be an increasing sequence of closed subspaces, and $$\mathcal{M}$$ be the closure of $$\mathcal{M}_1 \cup \mathcal{M}_2 \cup ...$$. If $$\pi_\mathcal{M}$$ is an orthogonal projection onto $$\mathcal{M}$$, then for every $$x \in \mathcal{M}$$ $$\pi_{\mathcal{M}_n}(x) \to x$$.''

Proof: Let $$\mathcal{N} = \{ x \in \mathcal{M}; \pi_{\mathcal{M}_n}(x) \to x (n \to \infty) \}$$. Then $$\mathcal{N}$$ is closed. Indeed, if $$x_j \in \mathcal{N}$$ and $$x_j \to x$$, then
 * $$\|\pi_{\mathcal{M}_n}(x) - x \| \le 2\|x - x_j\| + \|\pi_{\mathcal{M}_n}(x_j) - x_j \|$$

and so $$x \in \mathcal{N}$$. Since $$\mathcal{M} \subset \overline{\mathcal{N}}$$, the proof is complete. $$\square$$

Let $$(\mathfrak{H}_j, \|\cdot\|_j = \langle \cdot, \cdot \rangle_j)$$ be Hilbert spaces. The direct sum of $$\mathfrak{H}_1 \oplus \mathfrak{H}_2$$ is defined as follows: let $$\mathfrak{H}_1 \oplus \mathfrak{H}_2 = \{ (x_1, x_2); x_1 \in \mathfrak{H}_1, x_2 \in \mathfrak{H}_2 \} $$ and define
 * $$\langle x_1 \oplus x_2, y_1 \oplus y_2 \rangle = \langle x_1, y_1 \rangle_1 + \langle x_2, y_2 \rangle_2$$.

It is then easy to verify that $$(\mathfrak{H}_1 \oplus \mathfrak{H}_2, \langle \cdot, \cdot \rangle)$$ is a Hilbert space. It is also clear that this definition generalizes to a finite direct sum of Hilbert spaces. (For an infinite direct sum of Hilbert spaces, see Chapter 5.)

Recall from the previous chapter that an isometric surjection between Banach spaces is called "unitary".

3 Lemma (Hilbert adjoint) ''Define $$V: \mathfrak{H}_1 \oplus \mathfrak{H}_2 \to \mathfrak{H}_2 \oplus \mathfrak{H}_1$$ by $$V(x_1 \oplus x_2) = -x_2 \oplus x_1$$. (Clearly, $$V$$ is a unitary operator.) Then $$(V\operatorname{gra}T)^\bot$$ is a graph (of some linear operator) if and only if $$T$$ is densely defined.'' Proof: Set $$\mathcal{M} = (V\operatorname{gra}T)^\bot$$. Let $$u \in (\operatorname{dom}T^*)^\bot$$. Then
 * $$0 = \langle 0, -Tv \rangle_2 + \langle u, v \rangle_2 = \langle 0 \oplus u, -Tv \oplus v \rangle$$ for every $$v$$.

That is to say, $$0 \oplus u \in \mathcal{M}$$, which is a graph of a linear operator by assumption. Thus, $$u = 0$$. For the converse, suppose $$f \oplus u_1, f \oplus u_2 \in \mathcal{M}$$. Then
 * $$0 = \langle f \oplus u_j, -Tv \oplus v \rangle = \langle f, -Tv \rangle_2 + \langle u_j, v \rangle_1 \qquad $$ $$(j = 1, 2)$$

and so $$\langle u_1 - u_2, v \rangle_1 = 0$$ for every $$v$$ in the domain of $$T$$, dense. Thus, $$u_1 = u_2$$, and $$\mathcal{M}$$ is a graph of a function, say, $$S$$. The linear of $$S$$ can be checked in the similar manner.$$\square$$

Remark: In the proof of the lemma, the linear of $$T$$ was never used.

For a densely defined $$T$$, we thus obtained a linear operator which we call $$T^*$$. It is characterized uniquely by:
 * $$0 = \langle f, -Tu \rangle_2 + \langle T^*f, u \rangle_1 = \langle f \oplus T^*f, V(u \oplus Tu) \rangle$$ for every $$u$$,

or, more commonly,
 * $$\langle Tu, f \rangle_2 = \langle u, T^*f \rangle_1$$ for every $$u$$.

Furthermore, $$T^*f$$ is defined if and only if
 * $$u \mapsto \langle Tu, f \rangle$$

is continuous for every $$u \in \operatorname{dom}T$$. The operator $$T^*$$ is called the Hilbert adjoint (or just adjoint) of $$T$$. If $$T$$ is closed in addition to having dense domain, then
 * $$(V'\operatorname{gra}T^*)^\bot = (V' (V\operatorname{gra}T)^\bot)^\bot = \operatorname{gra}T^{\bot\bot} = \operatorname{gra}T$$

Here, $$V'(x_2, x_1) = -x_1 \oplus x_2$$. By the above lemma, $$T^*$$ is densely defined. More generally, if a densely defined operator $$T$$ has a closed extension $$S$$ (i.e., $$\operatorname{gra}T \subset \operatorname{gra}S = \overline{\operatorname{gra}S}$$), then $$S$$ and $$S^*$$ are both densely defined. It follows: $$\operatorname{gra}S^* \subset \operatorname{gra}T^*$$. That is, $$T^*$$ is densely defined and $$T^{**}$$ exists. That $$S = T^{**}$$ follows from the next theorem.

3 Theorem ''Let $$T: \mathfrak{H}_1 \to \mathfrak{H}_2$$ be a densely defined operator. If $$T^*$$ is also densely defined, then
 * $$\overline{\operatorname{gra}T} = \operatorname{gra}T^{**} = \operatorname{gra}S$$

for any closed extension $$S$$ of $$T$$.

Proof: As above,
 * $$(V'\operatorname{gra}T^*)^\bot = \operatorname{gra}T^{\bot\bot}$$

Here, the left-hand side is a graph of $$T^{**} $$. For the second identity, since $$\operatorname{gra}S$$ is a Hilbert space, it suffices to show $$ \operatorname{gra}T^\bot \cap \operatorname{gra}S = \{0\} $$. But this follows from Lemma 3.something.$$\square$$

The next corollary is obvious but is important in application.

3 Corollary ''Let $$\mathfrak{H}_1, \mathfrak{H}_2$$ be Hilbert spaces, and $$T: \mathfrak{H}_1 \to \mathfrak{H}_2$$ a closed densely defined linear operator. Then $$u \in \operatorname{dom}T$$ if and only if there is some $$K > 0$$ such that:
 * $$\|\langle T^*f, u \rangle \| \le K\|f\|$$ for every $$f \in \operatorname{dom}T^*$$

3 Lemma ''Let $$T: \mathfrak{H}_1 \to \mathfrak{H}_2$$ be a densely defined linear operator. Then $$\operatorname{ker}T^* = (\operatorname{ran}T)^\bot.$$''

Proof: $$f$$ is in either the left-hand side or the right-hand side if and only if:
 * $$0 = \langle T^*f, u \rangle = \langle f, Tu \rangle$$ for every $$u$$.

(Note that $$\langle f, Tu \rangle = 0$$ for every $$u$$ implies $$f \in \operatorname{dom}T^*$$.) $$\square$$

In particular, a closed densely defined operator has closed kernel. As an application we shall prove the next theorem.

3 Theorem ''Let $$T: \mathfrak{H}_1 \to \mathfrak{H}_2$$ be a closed densely defined linear operator. Then $$T$$ is surjective if and only if there is a $$K > 0$$ such that
 * $$\|f\|_2 \le K\|T^*f\|_1$$ for every $$f \in \operatorname{dom}T^*$$.''

Proof: Suppose $$T$$ is surjective. Since $$T$$ has closed range, it suffices to show the estimate for $$f \in (\operatorname{ker}T^*)^\bot = \operatorname{ran}T$$. Let $$u \in (\operatorname{ker}T)^\bot$$ with $$Tu = f$$. Denoting by $$G$$ the inverse of $$T$$ restricted to $$(\operatorname{ker}T)^\bot$$, we have:
 * $$\|f\|^2_2 \le \|T^*f\|_1 \|Gf\|_1 \le \|T^*f\| \|G\| \|f\|_2$$

The last inequality holds since $$G$$ is continuous by the closed graph theorem. To show the converse, let $$g \in \mathfrak{H}_2$$ be given. Since $$T^*$$ is injective, we can define a linear functional $$L$$ by $$L(T^*f) = \langle f, g \rangle_2$$ for $$f \in \mathfrak{H}_2$$.,
 * $$|L(T^*f)| = |\langle f, g \rangle_2| \le K\|T^*f\|$$ for every $$f \in \operatorname{dom}T^*$$.

Thus, $$L$$ is continuous on the range of $$T^*$$. It follows from the Hahn-Banach theorem that we may assume that $$L$$ is defined and continuous on $$\mathfrak{H}_1$$. Thus, by Theorem 3.something, we can write $$L(\cdot) = \langle \cdot, u \rangle_1$$ in $$\mathfrak{H}_1$$ with some $$u$$. Since $$L(T^*f)$$ is continuous for $$f \in \operatorname{dom}T^*$$,
 * $$L(T^*f) = \langle f, g \rangle_2 = \langle T^*f, u \rangle_1 = \langle f, T^{**}u \rangle_2$$ for every $$f \in \operatorname{dom}T^*$$.

Hence, $$Tu = T^{**}u = g$$. $$\square$$

3 Corollary ''Let $$T, \mathfrak{H}_1, \mathfrak{H}_2$$ be as given in the preceding theorem. Then $$\operatorname{ran}T$$ is closed if and only if $$\operatorname{ran}T^*$$ is closed.''

Proof: Define $$S: \mathfrak{H}_1 \to \operatorname{ran}T$$ by $$S = T$$. It thus suffices to show $$S^*$$ is surjective when $$T$$ has closed range (or equivalently $$S$$ is surjective.) Suppose $$S^*f_j$$ is convergent. The preceding theorem gives:
 * $$\|f_j - f_k\|_2 \le K\|S^*(f_j - f_k)\|_1 \to 0$$ as $$j, k \to \infty$$.

Thus, $$f_j \oplus S^*f_j$$ is Cauchy in the graph of $$S^*$$, which is closed. Hence, $$S^*f_j$$ converges within the range of $$S^*$$. The converse holds since $$T^{**} = T$$. $$\square$$

We shall now consider some concrete examples of densely defined linear operators.

3 Theorem ''$$T:\mathfrak{H}_1 \to \mathfrak{H}_2$$ is continuous if and only if $$T^*$$ is continuous. Moreover, when $$T$$ is continuous,
 * $$\|T\|^2 = \|T^*T\| = \|TT^*\| = \|T^*\|^2$$.''

Proof: It is clear that $$T^*$$ is defined everywhere, and its continuity is a consequence of the closed graph theorem. Conversely, if $$T^*$$ is continuous, then $$T^{**}$$ is continuous and $$T = T^{**}$$. For the second part,
 * $$\| T^*f \|^2_1 = | \langle T T^*f, f \rangle | \le \|T\| \|T^*f\|_2 \|f\|_2$$ for every $$f$$.

Thus, $$T^*$$ is continuous with $$\|T^*\| \le \|T\|$$. In particular, $$T^*T$$ is continuous, and so:
 * $$\| T^*f \|^2_1 \le \|T T^*\| \|f\|_2^2$$ for every $$f$$.

That is to say, $$\|T^*\|^2 \le \|T T^*\| \le \|T\|^2$$. Applying this result to $$T^*$$ in place of $$T$$ completes the proof.

The identity in the theorem shows that $$B(\mathcal{H})$$ is a $$C^*$$-algebra, which is a topic in Chapter 6.

3 Lemma ''Let $$S, T \in B(\mathfrak H)$$. If $$\langle Tx, x \rangle = \langle Sx, x \rangle$$ for $$x \in \mathfrak{H}$$, then $$S = T$$.

Proof: Let $$R = T - S$$. We have $$0 = \langle R(x + y), x + y \rangle = \langle Rx, y \rangle + \langle Ry, x \rangle$$ and $$0 = i\langle R(x + iy), x + iy \rangle = \langle Rx, y \rangle + i^2 \langle Ry, x \rangle$$. Summing the two we get: $$0 = 2\langle Rx, y \rangle$$ for $$x, y \in \mathfrak{H}$$. Taking $$y = Rx$$ gives $$0 = \|Rx\|^2$$ for all $$x \in \mathfrak{H}$$ or $$R = 0$$. $$\square$$

Remark: the above lemma is false if the underlying field is $$\mathbf{R}$$.

Recall that an isometric surjection is called unitary.

3 Corollary A linear operator $$U: \mathfrak{H}_1 \to \mathfrak{H}_2$$ is unitary if and only if $$U^*U$$ and $$UU^*$$ are identities.

Proof: Since $$(U^*Ux \mid x) = \|Ux\|^2 = (x \mid x)>$$, we see that $$U^*U$$ is the identity. Since $$UU^*U = U$$, $$UU^*$$ is the identity on the range of U, which is $$\mathcal{H}_2$$ by surjectivity. Conversely, since $$\|Ux\|^2_2 = \langle U^*Ux, x \rangle_1 = \|x\|^2_1$$, $$U$$ is an isometry. $$\square$$

Curiously, the hypothesis on linearity can be omitted:

3 Theorem If $$U: \mathcal{H}_1 \to \mathcal{H}_2$$ is a function'' such that
 * $$\| U(x) - U(y) \|_2 = \|x - y\|_1$$

for every x and y and $$U(0) = 0$$, then $$U$$ is a linear operator (and so unitary).''

Proof: Note that U is continuous. Since $$\|U(x)\| = \|U(x) - U(0)\| = \|x\|$$, we have:
 * $$\|x - y\|_1^2 = \|U(x) - U(y)\|_2^2 = \|x\|_1^2 - 2\operatorname{Re} ( U(x) \mid U(y) ) + \|y\|_1^2$$.

Thus,
 * $$\operatorname{Re}(x \mid y)_1 = \operatorname{Re}(U(x), U(y))_2$$

It now follows:
 * $$\|U(\alpha x + y) - \alpha U(x) - U(y)\|_2^2 = \|U(\alpha x + y) - U(\alpha x)\|_2^2 - 2\operatorname{Re} (y \mid y) + \|U(y)\|_2^2 = \|y\|_1^2 - 2\|y\|_1^2 + \|y\|_1^2 = 0$$

for any $$x, y \in \mathcal{H}_1$$ and scalar $$\alpha$$. $$\square$$

There is an analog of this result for Banach space. See, for example, http://www.helsinki.fi/~jvaisala/mazurulam.pdf)

3 Exercise ''Construct an example so as to show that an isometric operator (i.e., a linear operator that preserves norm) need not be unitary. (Hint: a shift operator.)''

A densely defined linear operator $$T$$ is called "symmetric" if $$\operatorname{gra}T \subset \operatorname{gra}T^*$$. If the equality in the above holds, then $$T$$ is called "self-adjoint". In light of Theorem 3.something, every self-adjoint is closed and densely defined. If $$T$$ is symmetric, then since $$T^{**}$$ is an extension of $$T$$,
 * $$\operatorname{gra}T \subset \operatorname{gra}T^* \cap \operatorname{gra}T^{**}$$.

3 Theorem ''Let $$T_j: \mathfrak{H}_j \to \mathfrak{H}_{j+1}$$ be densely defined linear operators for $$j = 1, 2$$. Then $$\operatorname{gra}T_1^* \circ T_2^* \subset \operatorname{gra}(T_2 \circ T_1)^*$$ where the equality holds if $$T_j^{**} = T_j$$ $$(j = 1, 2)$$ and $$T_1^* \circ T_2^*$$ is closed and densely defined.''

Proof: Let $$u \in \operatorname{dom}(T_1^* \circ T_2^*)$$. Then
 * $$\langle T_2 \circ T_1 v, u \rangle = \langle T_1 v, T_2^* u \rangle = \langle v, T_1^* \circ T_2^* u \rangle$$ for every $$v \in \operatorname{dom}(T_2 \circ T_1)$$.

But, by definition, $$(T_2 \circ T_1)^*u$$ denotes $$T_1^* \circ T_2^* u $$. Hence, $$(T_2 \circ T_1)^*$$ is an extension of $$T_1^* \circ T_2^*$$. For the second part, the fact we have just proved gives:
 * $$\operatorname{gra}T_1^* \circ T_2^* \subset \operatorname{gra}(T_2 \circ T_1)^* = \operatorname{gra}(T_2^{**} \circ T_1^{**})^* \subset \operatorname{gra}(T_1^* \circ T_2^*)^{**}$$. $$\square$$

3 Theorem ''Let $$T:\mathfrak{H}_1 \to \mathfrak{H}_2$$ be a Hilbert spaces. If $$T:\mathfrak{H}_1 \to \mathfrak{H}_2$$ is a closed densely defined operator, then $$T^*T$$ is a self-adjoint operator (in particular, densely defined and closed.)''

Proof: In light of the preceding theorem, it suffices to show that $$T^*T$$ is closed. Let $$u_j \in \operatorname{dom}T^*T$$ be a sequence such that $$(u_j, T^*T u_j)$$ converges to limit $$(u, v)$$. Since
 * $$\|T u_j - T u_k \|_2 \le 2 (\|T^*T (u_j - u_k) \|_1 + \|u_j - u_k\|_1)$$,

there is some $$f \in \mathfrak{H}_2$$ such that: $$\|Tu_j - f\|_2 \to 0$$. It follows from the closedness of $$T^*$$ that $$T^*f = v$$. Since $$\|u_j - u\|_1 + \|Tu_j - f\|_2 \to 0$$ and $$T$$ is closed, $$T^*Tu = T^*f = v$$. $$\square$$

3 Theorem ''Let $$T$$ be a symmetric densely defined operator. If $$T$$ is surjective, then $$T$$ is self-adjoint and injective and $$T^{-1}$$ is self-adjoint and bounded.''

Proof: If $$Tu = 0$$,
 * $$\langle Tu, v \rangle = \langle u, Tv \rangle$$ and $$u = 0$$

if $$T$$ has a dense range (for example, it is surjective). Thus, $$T$$ is injective. Since $$T^{-1}$$ is closed (by Lemma 2.something) and $$\operatorname{ran}T = \mathfrak{H}_2$$, $$T^{-1}: \mathfrak{H}_2 \to \operatorname{dom}T$$ is a continuous linear operator. Finally, we have:
 * $$\operatorname{gra}T^{-1} = V \operatorname{gra}T \subset V \operatorname{gra}T^* = \operatorname{gra}(T^*)^{-1} = \operatorname{gra}(T^{-1})^*$$.

Here, $$V(x_1 \oplus x_2) = x_2 \oplus x_1$$, and the equality holds since the domains of $$T$$ and $$T^*$$ coincide. Hence, $$T^{-1}$$ is self-adjoint. Since we have just proved that the inverse of a self-adjoint is self-adjoint, we have: $$(T^{-1})^{-1}$$ is self-adjoint.$$\square$$

3 Theorem ''Let $$\mathcal{M}$$ be a closed linear subspace of a Hilbert space $$\mathfrak{H}$$. Then $$\pi$$ is an orthogonal projection onto $$\mathcal{M}$$ if and only if $$\pi = \pi^* = \pi^2$$ and the range of $$\pi$$ is $$\mathcal{M}$$.''

Proof: The direct part is clear except for $$\pi = \pi^*$$. But we have:
 * $$\langle \pi(x), x \rangle = \|\pi(x)\|^2$$

since $$\pi(x)$$ and $$x - \pi(x)$$ are orthogonal. Thus, $$\pi$$ is real and so self-adjoint then. For the converse, we only have to verify $$x - \pi(x) \in \mathcal{M}^\bot$$ for every $$x$$. But we have: $$\pi (x - \pi(x)) = 0$$ and $$\operatorname{ker} (\pi) = \operatorname{ker} (\pi^*) = (\operatorname{ran} (\pi))^\bot = \mathcal{M}^\bot$$. $$\square$$

We shall now turn our attention to the spectral decomposition of a compact self-adjoint operator. Let $$T:\mathfrak{H} \to \mathfrak{H}$$ be a compact operator.

title: Functional Analysis/Hilbert spaces


text:
Methods Manual for Salt Lake Studies

Contents

 * 1) /Scope and Introduction/
 * 2) /Describing your lake - hydrogeomorphology/
 * 3) /Effects of salinity on evaporation rate/
 * 4) /Sequential crystallisation of brine salts/
 * 5) Measuring physical characteristics of the geomorphology
 * 6) /Measuring the hardness of playa crusts/
 * 7) Measuring physical characteristics of the brine
 * 8) /Salinity/
 * 9) Measuring brine density
 * 10) Measuring electrical conductivity
 * 11) Measuring refractive index
 * 12) Measuring chloride concentration
 * 13) Evaporating brine and weighing the residue (gravimetry)
 * 14) /Light penetration/
 * 15) /Color/
 * 16) /Turbidity/
 * 17) /Suspended solids/
 * 18) /Temperature/
 * 19) /pH/
 * 20) /Looking deeper - preparing to analyse/
 * 21) Inorganic chemical characteristics
 * 22) /Dissolved oxygen/
 * 23) Nutrients
 * 24) /Nitrogen/
 * 25) /Phosphorus/
 * 26) /Other nutrients/
 * 27) /Alkalinity/
 * 28) /Additional non-metallic substances/
 * 29) /Hardness/ (Calcium & Magnesium)
 * 30) /Additional metals/
 * 31) Organic chemical characteristics
 * 32) /BOD/
 * 33) /TOC/
 * 34) /Tannins and humic compounds/
 * 35) /Biological characteristics/
 * 36) /Emergent macrophyte vegetation and terrestrial fringing vegetation/
 * 37) /Submerged aquatic macrophytes and macroalgae/
 * 38) /Phytoplankton/
 * 39) /Periphyton/
 * 40) /Benthic mat communities/
 * 41) /Productivity/
 * 42) /Zooplankton/
 * 43) /Macroinvertebrates/
 * 44) /Vertebrates/
 * 45) /Bacteriology/
 * 46) /Putting it together - integrated biological approaches/
 * 47) /References/
 * 48) /Authors/

title: Methods Manual for Salt Lake Studies


text:
Card Games/Fool's Paradise

Fool's Paradise is an accumulation card game where certain cards have different point values and each player should try to get as many points as possible. This card game is best played with 4 to 5 players. At least 2 players are required and there is no upper limit, however as more players are playing in the same game, the game becomes less interesting and too slow.

The Deal
Given a dealer (one of the players), the dealer will deal out one card to each player face down plus one card to the center face up a total of five times. All cards dealt to the center will be placed in a horizontal row (still face up). Players receiving their cards may look at their cards as soon as they are dealt (as they are being dealt). The remainder of the deck is to be place in the center face down, these cards will become the draw pile. Once each player has five cards and there are five cards face up in the center, the game begins.

The Game
The player to the left of the dealer starts first.

A Player's Turn
When it is a player's turn, the player may collect cards according to the rules (read the next section). Once the player has collected his cards (if the player is able to), the player must draw one card from the draw pile into his or her hand. Upon drawing a card, the player's turn is over and the next player to the left may being his or her turn.

Collecting Cards
Collecting cards is the essence of this game. There are some complexities to collecting cards, so be sure to pay attention.

To collect a card or multiple cards, a player must first choose a card from his or her hand, and then find a card or multiple cards that sum up to a multiple of the chosen card's value, where each single card collected is less than or equal to the player's card. Card suit does not play any role in collecting cards, only in point value (discussed later).

This is best explained with an example:

Given a players hand: 2, 3, 9, K, A (of any suits) Given the center displays: J, 2, K, 5, 6 (of any suits)

The player could play a 2 and collect a 2 The player could play a K and collect a K The player could play a K and collect a K + (J, 2) The player could play a K and collect a K + (5, 6, 2)

In the example above, here are the 4 plays explained:
 * 1) Player plays a 2, and can collect any cards that sum to a multiple of 2. Only the center 2 can be collected. The 6 cannot be collected, even though it is a multiple of 2 because the 6 is greater than 2 and no card collected can be greater than the player's card (the 2).
 * 2) The player plays a king and collects a king from the center. This would be a waste since a king is valued at 13 and other cards can be summed to create 13.
 * 3) The player plays a king and collects the king, plus the jack and the two. Since the jack and the two together make 13, they can be collected with the king.
 * 4) Finally, the most efficient play, the king collects the king, along with the 5, 6, and 2 which sum to 13 as well. This play would acquire 4 cards from the center leaving only the jack.

All cards collected by a player are placed face down in a pile next to the player.

Not Collecting Cards
There are two instances where cards will not be collected on a player's turn.

The first is if the player cannot use any of his or her cards to collect any cards in the center. If this happens, the player must choose one card from his or her hand and lay it down in the center, then subsequently draw a card from the draw pile.

The second is if the player wishes to leave cards that would otherwise be collected in the center to be collected next round when it is the player's turn again. This can be beneficial since it allows for the player to clear the board. However, the player also opens up the opportunity for another player to steal the cards that he has left. This move is the Fool's Paradise.

Clearing the board
When a player collects the final card that is face up in the center, the player is said to have cleared the board. When this occurs, after the player collects his card from the draw pile, the dealer must deal out five cards face up to the center.

Every time a player clears the board, the player is awarded one point immediately.

Ending
The game ends when all players are without cards. The player to play the last card, whether or not he or she collects any cards with the last card, is given all remaining cards in the center. After the game ends, collected cards are inspected and counted for points.

Card Values

 * Two through ten is face value
 * Ace may be one or fourteen
 * Jack is eleven
 * Queen is twelve
 * King is thirteen

Points
At the end of a game, everyone must count how many points they have. Here is the break down for how to count your points.

 * Spades: Any spade is worth one point.
 * Twos: Each two is worth two points.
 * Jack of Clubs: The Jack of Clubs is worth 5 points.
 * Most Cards Collected: The player who collects the most cards during the round is awarded one point.

NOTE: the Two of Spades is actually worth 3 points, one for Spades and two for a Two

Winning
To win at Fool's Paradise, you have to be the first player to reach a certain number of points. This number should be decided upon before the game starts. A good normal number to play with is 50 points.

title: Card Games/Fool's Paradise


text:
Real Analysis/Foreword

What is Analysis?
Prof. Elliot Lieb of Princeton University defines analysis as the "art of taking limits", and further adds that "estimates are the heart and soul of analysis". Mathematics is often roughly subdivided into analysis, algebra and topology, so the coverage of each of these fields is quite broad. This book is concerned in particular with analysis in the context of the real numbers — there are many other fields of analysis, such as complex analysis, functional analysis and harmonic analysis. It will first develop the basic concepts needed for the idea of functions, then move on to the more analysis-based topics.

Analysis or Calculus?
Analysis is concerned with primarily the same topics as Calculus, such as limits, derivatives, and integrals, but in a mathematical way rather than in a simply practical way. Before you study analysis, you may want to study calculus; you will end up repeating much of the same material when you come back to analysis, but you will understand its practical significance. It may seem like a wasteful duplication of effort, but you will feel much more comfortable with many of the basic concepts of analysis.

On the other hand, when studying calculus you may be dismayed at the frequent statement of rules for performing various operations with little or no justification. The study of analysis puts all these on a formal basis and provides that justification.

Preliminaries
In much of analysis, arguments must be constructed very carefully, and it must be possible to make statements very precisely. To this end, it is important to be familiar with the notation of mathematical logic, in particular the 'for all' ( $$\forall$$ ) and 'there exists' ( $$\exists$$ ) notations. This notation will be used in definitions and proofs throughout the book, and it is essential to understand what these symbols mean and how they relate to each other before attempting to understand the material presented.

The core of real analysis uses very little from other areas of mathematics. Some general fundamentals that will be useful include Set Theory, especially the sections on infinite sets and cardinality. In many places, but particularly for work on Sequences, an understanding of induction and recursion (on $$\mathbb N$$) is important; however, nothing will be required that goes beyond a standard introduction to proof technique. For some of the more advanced topics, some knowledge of Topology may be helpful. Familiarity with the concepts of Calculus will make this book easier to get through, but not necessary as all relevant concepts will be defined in the text when needed.

Note also that some minimal background in Linear Algebra is assumed in the chapter on Multivariable analysis, but this is optional as no other chapter is dependent on this one.

Advice for Readers
Like all math Wikibooks, Real Analysis is encyclopedic. Unlike most print books, it has not been designed to supplement any course or instruction. As such, it contains several ideas and concepts stacked together which might baffle the uninitiated reader. We plan to have a major reorganisation of the contents and also to provide a guide to readers so as to make the best use of this book. Also a major problem with this book in particular is the lack of comprehensive exercises, especially in the later chapters.

Editors are invited to contribute towards any of these improvements.

School of Mathematics

title: Real Analysis/Foreword


text:
Descriptive Religion/Describing world religions

Describing World Religions

Religion is a multifaceted subject area. Religions differ in their histories, what their adherents believe, their geographic range and statistical characteristics, their art and literature, philosophies, medical traditions, practices, and so on. How would you go about describing religions?

It would have something to do with why you want to describe the religion. Two of the three resources I've given below approach the issue from a position of understanding other religions for the purpose of effectively evangelizing them. I won't be doing that here and the reason that I included those two resources is that they happen to be accurate and informative. Instead of emphasizing the negative characteristics of the religions (as a similar work might have done, say, a century ago), they strive to be accurate.

I intend to proceed systematically and will include the following sections under each religion:

Names and meanings of the names: There is often confusion related to the names of a religion. For instance, what does "Jewish" mean. Is it the name of a race, a nationality, or a religion?

Adherents: What is the geographic range and makeup of the adherents of each religion at the time of posting?

History: When, where, and by whom was the religion founded and who did it become as it is now? Religions are dynamic systems and almost never stand still over time.

Major scriptures. Most of the sources of world religions are available on the Internet for free so that it's easy to find out specifically what a religion says about any particular thing - at least, as far as the orthodox position.

Denominations: Few religions come in only one flavor. I will describe the major branches of each religion.

Specific beliefs: What do the adherents of each religion thing about God, Humanity, Animals, the primary problems of Humanity and their solutions, the spirit, death, and the afterlife, morality, proselytizing, and society.

Glossary: Finally, I will present a glossary of terms specific to each religion.

The major religions appeared over a period of 3000 years - from 1500 BC to 1500 AD. Two periods of time saw the appearances of many religions. In the Axial Age, between 800 and 200 BC, Buddhism, Taoism, Shinto, and Confucianism were founded along with Greek philosophy and the rise of the major prophets in Israel. Then again, the last two centuries saw a proliferation of new religions and return of ancient religions long dead.

Major milestones were:

c. 1500 BC The rise of Hinduism in the area of present day India. c. 1440 BC Proposed date of the Exodus, the initial event in both Judaism and Jewish nationalism c. 660 BC reign of Jimmu Tenno, the traditional beginning on Shinto c. 600 BC The life of Lao Tze, traditional founder of Taoism 563 BC Traditional date of the birth of the Buddha, Siddhartha Gautama 551 BC Traditional date of the birth of Confucius c. 30 The crucifixion of Jesus 622 Mohammad arrived in Medina 1440 Birth of the Indian mystic Kabir, founder of Sikhism

Objects of art, as forms of communication of deeper meaning than is possible with technical writing and speech, are important in most religion. Inspired writing (writing thought to be produced under the direct guidance of deity) form the scriptures of the great religions, but creeds, commentaries, devotional literature, and educational materials are also important. Most religions also have a hymnology of music which conveys ideas central to the religions. Added to those, relics, architecture of holy places, recorded sermons form a rich heritage of the religions.

Few religions exist in a pure form - most have branches resulting from internal disagreements and variational interpretations of scriptures and denominations. There are thousands of denominations today but we will look at the major branches of the world religions.

World religions form a kaleidoscopic tapestry of tradition and society that would (and does) fill encyclopedias. Hopefully, this work will give more than a superficial introduction.

Gaer, Joseph (1963) What the great Religions Believe. The New American Library, Inc.: New York, NY

McDowell, Josh, & Don Stewart (1982) Understanding Non-Christian Religions. Here's Life Publishers, Inc.: San Bernardino, CA

Halverson, Dean C. (gen. ed.) (1996) The Compact guide to World Religions. Bethany House Publishers: Minneapolis, MN


text:
Descriptive Religion/Religions, denominations, sects, and cults

Religions, denominations, sects, and cults

Before we begin describing the individual religions, we need to be clear on a few general terms. "Religion", "denomination", "sect", "cult" - these terms do not have commonly agreed upon meanings. The best we can do when using them is to define what we mean before hand, so let me do that here.

I will use the definition already in use in the other books of this series for "religion". Religion is "the study and/or practice of what is generally considered to be religion - personal or organized." (From Theology 1: Philosophy of Religion - What is religion?). An adherent is a person who accepts the basic tenets of a religion. Every adherent has a slightly (or much more than slightly) different take on their religion and that makes for a great deal of complexity when talking about specific religions. For instance, I can say that all Catholics believe that the wine of the Eucharist is changed into the blood of Christ but I'm sure that a Catholic can be found that doesn't believe this. Just keep in mind that, when I talk about religions, I am overgeneralizing by necessity and am sticking to the main "party line" while describing some (but by no means all) major variations.

An orthodox belief is a belief that is supported by the primary tradition of a religion. A heterodox belief is one that is not. A heresy is a heterodoxy that is condemned by the authoritative clergy of a religion.

The word "creed" is derived from the Latin "credo" which means "I believe". A creed, then, is an authoritative statement of the central beliefs in a religion. A doctrine is a body of beliefs about a particular topic in a religion - particularly a collections of concepts which are actively taught within that religion. A dogma is a belief that is so established in a religion that it is accepted without question.

Almost all religions have variations that result from different interpretations of fundamental scriptures, traditions, or teachings. those variations are called "denominations". They are "spin-offs" of a parent religion. The term "sect" will apply to a denomination that is considered by the parent religion to be extreme or heretical but which becomes firmly established over time.

The term "cult" is difficult but important. It was originally a synonym of "denomination" but has evolved to have a darker usage. I do not want to use it in its most extreme sense of a dangerous or destructive body of religious beliefs but I will use it in its milder sense as a specific religion which is characterized by a central, authoritative, charismatic leader that exerts considerable power over the adherents of the religion.

Most of these definitions seem to be quite in line with what most people mean when they use the words. It's most important that you know what I mean when I use them.

The original religion among denominations may or may not exist currently but it can often be approximately described by original scriptures and by comparing commonalities among the denominations. For instance, much of Buddhism today is drastically different from the Buddha's original intention but there is a conceptual "core" Buddhism. Also, the range of variation in shamanism is vast but a core shamanism can be recognized by looking at the more "primitive" Siberian forms and by comparing the other forms.

The term "core" does not seem to be commonly used but it will come in handy here and I will use it occasionally. Also see it's use in the reference cited below.

Connelly, Paul (1996) Definition of Religion and Related Terms http://www.darc.org/connelly/religion1.html


text:
User:Carlo.milanesi

See Carlo Milanesi Home page

I am Italian and I live in Italy.

I wrote a book about computer programming in Italian, and on the May 3rd 2008 I published it in my Web site: C++ Ottimizzato.

On May 11th 2008 I published it as the Italian wikibook Ottimizzare C++.

On May 24th 2008 I started to translate it from Italian into the English wikibook Optimizing C++.

title: User:Carlo.milanesi


text:
Optimizing C++

Audience
This book is for intermediate C++ programmers (i.e. studying C++ for at least one year) who want to improve the performance of the software they develop.

Subject
This book contains guidelines and advices on how to write efficient software using the C++ language. Software correctness and maintainability are taken into account, but are not the primary concerns of the guidelines.

Most of the guidelines presented are not platform specific, and so are applicable to every processor, to every compiler, and to every operating system; the few platform specific advices try to cover most platforms.

Table of Contents

 * 1) Introduction
 * 2) Optimization life cycle
 * 3) Writing efficient code
 * 4) Performance improving features
 * 5) Performance worsening features
 * 6) Constructions and destructions
 * 7) Allocations and deallocations
 * 8) Memory access
 * 9) Thread usage
 * 10) General optimization techniques
 * 11) Input/Output
 * 12) Memoization
 * 13) Sorting
 * 14) Other techniques
 * 15) Code optimization
 * 16) Allocations and deallocations
 * 17) Run-time support
 * 18) Instruction count
 * 19) Constructions and destructions
 * 20) Pipeline
 * 21) Memory access
 * 22) Faster operations
 * 23) Compile time optimization
 * 24) Tools
 * 25) Further reading

Guidelines for editors

Ottimizzare C++

title: Optimizing C++


text:
Optimizing C++/Introduction

One of the main reason for preferring C++ over simpler, higher-level programming languages is that C++ allows the development of complex software in a way that makes more efficient use of hardware resources than when using these other languages. The language does not guarantee efficient code automatically, but provides a toolchest that aids programmers in the pursuit of efficiency. Sloppy C++ code may be no more efficient than higher-level implementations of the same algorithms, but a good C++ programmer with knowledge of the language can write software that is efficient from the first cut and then optimize the code further. This book provides techniques for writing efficient code and for optimizing existing software.

Often, there is no single solution to a programming problem that is optimal for all cases. Thus, optimization generally does not mean writing optimally performing software; rather, it means incrementally changing (refactoring) software to increase its performance, bringing it closer to the optimum.

Such optimization requires that the software source is written in a sufficiently modular way that performance critical parts can be isolated. With suitably written code, it then requires the use of tools, libraries, knowledge and time to change those parts in a way that increases the overall execution speed of the software.

Nowadays, many optimizations are already performed by compilers and are no longer the programmer's burden. This book discusses higher-level optimizations that present-day compilers are not (yet) able to perform.

This book is aimed at readers already familiar with the C++ language, who want to use it to develop high performance application software or software libraries.

Almost all of the optimization techniques presented are platform independent and, therefore, there will be few references to specific operating systems, processor architectures or compilers. However, some of the presented techniques may be ineffective or inapplicable in some combinations of operating system/processor/compiler.

Ottimizzare C++/Introduzione

title: Optimizing C++/Introduction


text:
Sewing

Table of Contents

 * /Introduction/
 * /Fabric/
 * /Hand sewing/
 * /Sewing utensils/
 * /Seams/
 * /Stitches/
 * /Embroidery/

title: Sewing


text: Template:Optimizing C++

title: Template:Optimizing C++