text:
Dill

Dill (Anethum graveolens) is an annual herb in the celery family Apiaceae. It is native to North Africa, Iran, and the Arabian Peninsula; it is grown widely in Eurasia, where its leaves and seeds are used as a herb or spice for flavouring food.

Etymology
The word dill and its close relatives are found in most of the Germanic languages; its ultimate origin is unknown.

Taxonomy
The genus name Anethum is the Latin form of Greek ἄνῑσον / ἄνησον / ἄνηθον / ἄνητον, which meant both "dill" and "anise". The form 'anīsum' came to be used for anise, and 'anēthum' for dill. The Latin word is the origin of dill's names in the Western Romance languages ('anet', 'aneldo' etc.), and also of the obsolete English 'anet'.

Botany
Dill grows up to 1.5 - 5 ft from a taproot like a carrot. Its stems are slender and hollow with finely divided, softly delicate leaves; the leaves are alternately arranged, 10 - 20 cm long with ultimate leaf divisions are 1 - 2 mm broad, slightly broader than the similar leaves of fennel, which are threadlike, less than 1 mm broad, but harder in texture.

In hot or dry weather, small white to yellow scented flowers form in small umbels 1 - 3+1/2 in diameter from one long stalk. The seeds come from dried up fruit 4 - 5 mm long and 1 mm thick, and straight to slightly curved with a longitudinally ridged surface.

Cultivation
Successful cultivation requires warm to hot summers with high sunshine levels; even partial shade will reduce the yield substantially. It also prefers rich, well-drained soil. The seed is harvested by cutting the flower heads off the stalks when the seed is beginning to ripen. The seed heads are placed upside down in a paper bag and left in a warm, dry place for a week. The seeds then separate from the stems easily for storage in an airtight container.

These plants, like their fennel and parsley relatives, often are eaten by black swallowtail caterpillars in areas where that species occurs. For this reason, they may be included in some butterfly gardens.

History
Dill has been found in the tomb of Egyptian Pharaoh Amenhotep II, dating to around 1400 BC. It was also later found in the Greek city of Samos, around the 7th century BC, and mentioned in the writings of Theophrastus (371–287 BC). In Greek mythology, the dill was originally a young man named Anethus who was transformed into the plant.

Aroma profile

 * Apiole and dillapiole
 * Carvone
 * Limonene
 * Myristicin
 * Umbelliferone

Fresh and dried dill leaves (sometimes called "dill weed" or "dillweed" to distinguish it from dill seed) are widely used as herbs in Europe and in central and south-eastern Asia.

Like caraway, the fern-like leaves of dill are aromatic and are used to flavour many foods such as gravlax (cured salmon) and other fish dishes, borscht, and other soups, as well as pickles (where the dill flower is sometimes used). Dill is best when used fresh, as it loses its flavor rapidly if dried. However, freeze-dried dill leaves retain their flavour relatively well for a few months.

Dill oil is extracted from the leaves, stems, and seeds of the plant. The oil from the seeds is distilled and used in the manufacturing of soaps.

Dill is the eponymous ingredient in dill pickles.

Central and eastern Europe
In central and eastern Europe, the Nordic countries, the Baltic states, Ukraine, and Russia, dill is a staple culinary herb along with chives and parsley. Fresh, finely cut dill leaves are used as a topping in soups, especially the hot red borsht and the cold borsht mixed with curds, kefir, yogurt, or sour cream, which is served during hot summer weather and is called 'okroshka'. It also is popular in summer to drink fermented milk (curds, kefir, yogurt, or buttermilk) mixed with dill (and sometimes other herbs).

In the same way, dill is used as a topping for boiled potatoes covered with fresh butter – especially in summer when there are so-called "new", or young, potatoes. The dill leaves may be mixed with butter, making a dill butter, to serve the same purpose. Dill leaves mixed with tvorog form one of the traditional cheese spreads used for sandwiches. Fresh dill leaves are used throughout the year as an ingredient in salads, e.g., one made of lettuce, fresh cucumbers, and tomatoes, as basil leaves are used in Italy and Greece.

Russian cuisine is noted for liberal use of dill, where it is known as укроп. It is supposed to have antiflatulent properties; some Russian cosmonauts recommended its use in human spaceflight due to such properties being beneficial in confined quarters with a closed air supply.

In Polish cuisine, fresh dill leaves mixed with sour cream are the basis for dressings. It is especially popular to use this kind of sauce with freshly cut cucumbers, which are almost wholly immersed in the sauce, making a salad called 'mizeria'. Dill sauce is used hot for baked freshwater fish and for chicken or turkey breast, or used hot or cold for hard-boiled eggs. A dill-based soup, (zupa koperkowa), served with potatoes and hard-boiled eggs, is popular in Poland. Whole stems including roots and flower buds are used traditionally to prepare Polish-style pickled cucumbers (ogórki kiszone), especially the so-called low-salt cucumbers (ogórki małosolne). Whole stems of dill (often including the roots) also are cooked with potatoes, especially the potatoes of autumn and winter, so they resemble the flavour of the newer potatoes found in summer. Some kinds of fish, especially trout and salmon, traditionally are baked with the stems and leaves of dill.

In the Czech Republic, white dill sauce made of cream (or milk), butter, flour, vinegar, and dill is called 'koprová omáčka' (also 'koprovka' or 'kopračka') and is served either with boiled eggs and potatoes, or with dumplings and boiled beef. Another Czech dish with dill is a soup called 'kulajda' that contains mushrooms (traditionally wild ones).

In Germany, dill is popular as a seasoning for fish and many other dishes, chopped as a garnish on potatoes, and as a flavouring in pickles.

In the UK, dill may be used in fish pie.

In Bulgaria dill is widely used in traditional vegetable salads, and most notably the yogurt-based cold soup Tarator. It is also used in the preparation of sour pickles, cabbage, and other dishes.

In Romania dill (mărar) is widely used as an ingredient for soups such as 'borş' (pronounced "borsh"), pickles, and other dishes, especially those based on peas, beans, and cabbage. It is popular for dishes based on potatoes and mushrooms and may be found in many summer salads (especially cucumber salad, cabbage salad and lettuce salad). During springtime, it is used in omelets with spring onions. It often complements sauces based on sour cream or yogurt and is mixed with salted cheese and used as a filling. Another popular dish with dill as a main ingredient is dill sauce, which is served with eggs and fried sausages.

In Hungary, dill is very widely used. It is popular as a sauce or filling, and mixed with a type of cottage cheese. Dill is also used for pickling and in salads. The Hungarian name for dill is 'kapor'.

In Serbia, dill is known as 'mirodjija' and is used as an addition to soups, potato and cucumber salads, and French fries. It features in the Serbian proverb, "бити мирођија у свакој чорби" /biti mirodjija u svakoj čorbi/ (to be a dill in every soup), which corresponds to the English proverb "to have a finger in every pie".

In Greece, dill is known as 'άνηθος' (anithos). In antiquity it was used as an ingredient in wines that were called "anithites oinos" (wine with anithos-dill). In modern days, dill is used in salads, soups, sauces, and fish and vegetable dishes.

In Santa Maria, Azores, dill (endro) is the most important ingredient of the traditional Holy Ghost soup (sopa do Espírito Santo). Dill is found ubiquitously in Santa Maria, yet, is rare in the other Azorean Islands.

In Sweden, dill is a common spice or herb. The flowers of fully grown dill are called 'krondill' (crown dill) and used when cooking crayfish. The krondill is put into the water after the crayfish is boiled, but still in hot and salt water. Then the entire dish is refrigerated for at least 24 hours before being served (with toasted bread and butter). Krondill is also used to flavor pickles and vodka. After a month or two of fermentation, the cucumber pickles are ready to eat, for instance, with pork, brown sauce, and potatoes, as a sweetener. The thinner part of dill and young plants may be used with boiled fresh potatoes (especially the first potatoes of the year, new potatoes, which usually are small and have a very thin skin). In salads it is used together with, or instead, of other green herbs, such as parsley, chives, and basil. It is often paired up with chives when used in food. Dill is often used to flavour fish and seafood in Sweden, for example, gravlax and various herring pickles, among them the traditional, 'sill i dill' (literally 'herring in dill'). In contrast to the various fish dishes flavoured with dill, there is also a traditional Swedish dish called, 'dillkött', which is a meaty stew flavoured with dill. The dish commonly contains pieces of veal or lamb that are boiled until tender and then served together with a vinegary dill sauce. Dill seeds may be used in breads or 'akvavit'. A newer, non-traditional use of dill is to pair it with chives as a flavouring for potato chips. These are called 'dillchips' and are quite popular in Sweden.

In Finland, the uses of dill are very similar to those in Sweden, including flavouring potato chips and, comparably less popularly, in a dish comparable to 'dillkött' ('tilliliha'). Generally, the use of dill in Finland, however, is not as extensive as in large parts of central and eastern Europe, particularly Russia but including even the ethnolinguistically close Estonia.

Asia and Middle East
In Iran, dill is known as 'shevid' and sometimes, is used with rice and called 'shevid-polo'. It also is used in Iranian 'aash' recipes, and similarly, is called sheved in Persian.

In India, dill is known as 'Sholpa' in Bengali, shepu (शेपू) in Marathi, sheppi (शेप्पी) in Konkani, savaa in Hindi, or soa in Punjabi. In Telugu, it is called 'Soa-kura' (herb greens). It also is called sabbasige soppu (ಸಬ್ಬಸಿಗೆ ಸೊಪ್ಪು) in Kannada. In Tamil it is known as sada kuppi (சதகுப்பி). In Malayalam, it is ചതകുപ്പ (chathakuppa) or ശതകുപ്പ (sathakuppa). In Sanskrit, this herb is called shatapushpa. In Gujarati, it is known as suva (સૂવા). In India, dill is prepared in the manner of yellow 'moong dal', as a main-course dish. It is considered to have very good antiflatulent properties, so it is used as 'mukhwas', or an after-meal digestive. Traditionally, it is given to mothers immediately after childbirth. In the state of Uttar Pradesh in India, a small amount of fresh dill is cooked along with cut potatoes and fresh fenugreek leaves (Hindi आलू-मेथी-सोया).

In Manipur, dill, locally known as pakhon, is an essential ingredient of chagem pomba – a traditional Manipuri dish made with fermented soybean and rice.

In Laos and parts of northern Thailand, dill is known in English as Lao coriander (ຜັກຊີ or ผักชีลาว), and served as a side with salad yum or papaya salad. In the Lao language, it is called 'phak see', and in Thai, it is known as 'phak chee Lao'. In Lao cuisine, Lao coriander is used extensively in traditional Lao dishes such as 'mok pa' (steamed fish in banana leaf) and several coconut milk curries that contain fish or prawns.

In China dill is called colloquially, 'huíxiāng' (茴香, perfume of Hui people), or more properly 'shíluó' (莳萝/蒔蘿). It is a common filling in 'baozi', 'jiaozi' and 'xianbing' and may be used as vegetarian with rice vermicelli, or combined with either meat or eggs. Vegetarian dill baozi are a common part of a Beijing breakfast. In baozi and xianbing, it often is interchangeable with non-bulbing fennel and the term 茴香 also may refer to fennel, similarly to caraway and coriander leaf, sharing a name in Chinese as well. Dill also may be stir fried as a potherb, often with egg, in the same manner as Chinese chives. In Northern China, Beijing, Inner-Mongolia, Ningxia, Gansu, and Xinjiang, dill seeds commonly are called 'zīrán' (孜然), but also 'kūmíng' (枯茗), 'kūmíngzi' (枯茗子), 'shíluózi' (莳萝子/蒔蘿子), 'xiǎohuíxiāngzi' (小茴香子) and are used with pepper for lamb meat. In the whole of China, 'yángchuàn' (羊串) or 'yángròu chuàn' (羊肉串), lamb brochette, a speciality from Uyghurs, uses cumin and pepper.

In Taiwan, it is also commonly used as a filling in steamed buns (baozi) and dumplings (jiaozi).

In Vietnam, the use of dill in cooking is regional. It is used mainly in northern Vietnamese cuisine.

Middle East
In Arab countries, dill seed, called ain jaradeh (grasshopper's eye), is used as a spice in cold dishes such as 'fattoush' and pickles. In Arab countries of the Persian Gulf, dill is called 'shibint' and is used mostly in fish dishes. In Egypt, dillweed is commonly used to flavour cabbage dishes, including 'mahshi koronb' (stuffed cabbage leaves). In Israel, dill weed is used in salads and also to flavour omelettes, often alongside parsley.

Companion planting
When used as a companion plant, dill attracts many beneficial insects as the umbrella flower heads go to seed. It makes a good companion plant for cucumbers and broccoli.

Tomatoes will benefit from dill when it is young since it will repel harmful pests while attracting pollinators, but must be pruned as it matures to prevent the dill from flowering, as this will slow or stop the growth of tomatoes.

title: Dill


text:
Dual space

In mathematics, any vector space $$V$$ has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on $$V,$$ together with the vector space structure of pointwise addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.

Early terms for dual include polarer Raum [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.

Algebraic dual space
Given any vector space $$V$$ over a field $$F$$, the (algebraic) dual space $$V^{*}$$ (alternatively denoted by $$V^{\lor}$$ or $$V'$$ ) is defined as the set of all linear maps $$\varphi: V \to F$$ (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted $$\hom (V, F)$$. The dual space $$V^*$$ itself becomes a vector space over $$F$$ when equipped with an addition and scalar multiplication satisfying:

\begin{align} (\varphi + \psi)(x) &= \varphi(x) + \psi(x) \\ (a \varphi)(x) &= a \left(\varphi(x)\right) \end{align}$$ for all $$\varphi, \psi \in V^*$$, $$x \in V$$, and $$a \in F$$.

Elements of the algebraic dual space $$V^*$$ are sometimes called covectors, one-forms, or linear forms.

The pairing of a functional $$\varphi$$ in the dual space $$V^*$$ and an element $$x$$ of $$V$$ is sometimes denoted by a bracket: $$\varphi (x) = [x, \varphi]$$ or $$\varphi (x) = \langle x, \varphi \rangle$$. This pairing defines a nondegenerate bilinear mapping $$\langle \cdot, \cdot \rangle : V \times V^* \to F$$ called the natural pairing.

Finite-dimensional case
If $$V$$ is finite-dimensional, then $$V^*$$ has the same dimension as $$V$$. Given a basis $$\{\mathbf{e}_1,\dots,\mathbf{e}_n\}$$ in $$V$$, it is possible to construct a specific basis in $$V^*$$, called the dual basis. This dual basis is a set $$\{\mathbf{e}^1,\dots,\mathbf{e}^n\}$$ of linear functionals on $$V$$, defined by the relation
 * $$ \mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n $$

for any choice of coefficients $$c^i\in F$$. In particular, letting in turn each one of those coefficients be equal to one and the other coefficients zero, gives the system of equations
 * $$ \mathbf{e}^i(\mathbf{e}_j) = \delta^{i}_{j} $$

where $$\delta^{i}_{j}$$ is the Kronecker delta symbol. This property is referred to as the bi-orthogonality property.

Consider $$\{\mathbf{e}_1,\dots,\mathbf{e}_n\}$$ the basis of V. Let $$\{\mathbf{e}^1,\dots,\mathbf{e}^n\}$$ be defined as the following:

$$ \mathbf{e}^i(c^1 \mathbf{e}_1+\cdots+c^n\mathbf{e}_n) = c^i, \quad i=1,\ldots,n $$.

These are a basis of $$V^*$$ because:

 * 1) The $$\mathbf{e}^i, i=1, 2, \dots, n, $$ are linear functionals, which map $$ x,y \in V $$ such as $$ x=  \alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n $$ and $$ y = \beta_1\mathbf{e}_1 + \dots + \beta_n \mathbf{e}_n $$ to scalars $$ \mathbf{e}^i(x)=\alpha_i $$ and $$ \mathbf{e}^i(y)=\beta_i$$. Then also, $$ x+\lambda y=(\alpha_1+\lambda \beta_1)\mathbf{e}_1 + \dots + (\alpha_n+\lambda\beta_n)\mathbf{e}_n $$ and $$ \mathbf{e}^i(x+\lambda y)=\alpha_i+\lambda\beta_i=\mathbf{e}^i(x)+\lambda \mathbf{e}^i(y) $$. Therefore, $$ \mathbf{e}^i  \in V^* $$ for $$ i= 1, 2, \dots, n $$.
 * 2) Suppose $$ \lambda_1 \mathbf{e}^1 + \cdots + \lambda_n \mathbf{e}^n =0  \in V^*$$. Applying this functional on the basis vectors of $$ V $$ successively, lead us to $$ \lambda_1=\lambda_2= \dots=\lambda_n=0 $$ (The functional applied in $$ \mathbf{e}_i $$ results in $$ \lambda_i $$). Therefore, $$\{\mathbf{e}^1,\dots,\mathbf{e}^n\}$$ is linearly independent on $$V^*$$.
 * 3) Lastly, consider $$ g \in V^* $$. Then

g(x)=g(\alpha_1\mathbf{e}_1 + \dots + \alpha_n\mathbf{e}_n)=\alpha_1g(\mathbf{e}_1) + \dots + \alpha_ng(\mathbf{e}_n)=\mathbf{e}^1(x)g(\mathbf{e}_1) + \dots + \mathbf{e}^n(x)g(\mathbf{e}_n) $$ and $$\{\mathbf{e}^1,\dots,\mathbf{e}^n\}$$ generates $$V^*$$. Hence, it is a basis of $$ V^*$$.

For example, if $$V$$ is $$\R^2$$, let its basis be chosen as $$\{\mathbf{e}_1=(1/2,1/2),\mathbf{e}_2=(0,1)\}$$. The basis vectors are not orthogonal to each other. Then, $$\mathbf{e}^1$$ and $$\mathbf{e}^2$$ are one-forms (functions that map a vector to a scalar) such that $$\mathbf{e}^1(\mathbf{e}_1)=1$$, $$\mathbf{e}^1(\mathbf{e}_2)=0$$, $$\mathbf{e}^2(\mathbf{e}_1)=0$$, and $$\mathbf{e}^2(\mathbf{e}_2)=1$$. (Note: The superscript here is the index, not an exponent.) This system of equations can be expressed using matrix notation as

\begin{bmatrix} e^{11} & e^{12} \\ e^{21} & e^{22} \end{bmatrix} \begin{bmatrix} e_{11} & e_{21} \\ e_{12} & e_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. $$ Solving for the unknown values in the first matrix shows the dual basis to be $$\{\mathbf{e}^1=(2,0),\mathbf{e}^2=(-1,1)\}$$. Because $$\mathbf{e}^1$$ and $$\mathbf{e}^2$$ are functionals, they can be rewritten as $$\mathbf{e}^1(x,y)=2x$$ and $$\mathbf{e}^2(x,y)=-x+y$$.

In general, when $$V$$ is $$\R^n$$, if $$E=[\mathbf{e}_1|\cdots|\mathbf{e}_n]$$ is a matrix whose columns are the basis vectors and $$\hat{E}=[\mathbf{e}^1|\cdots|\mathbf{e}^n]$$ is a matrix whose columns are the dual basis vectors, then
 * $$\hat{E}^\textrm{T}\cdot E = I_n,$$

where $$I_n$$ is the identity matrix of order $$n$$. The biorthogonality property of these two basis sets allows any point $$\mathbf{x}\in V$$ to be represented as
 * $$\mathbf{x} = \sum_i \langle\mathbf{x},\mathbf{e}^i \rangle \mathbf{e}_i = \sum_i \langle \mathbf{x}, \mathbf{e}_i \rangle \mathbf{e}^i,$$

even when the basis vectors are not orthogonal to each other. Strictly speaking, the above statement only makes sense once the inner product $$\langle \cdot, \cdot \rangle$$ and the corresponding duality pairing are introduced, as described below in .

In particular, $$\R^n$$ can be interpreted as the space of columns of $$n$$ real numbers, its dual space is typically written as the space of rows of $$n$$ real numbers. Such a row acts on $$\R^n$$ as a linear functional by ordinary matrix multiplication. This is because a functional maps every $$n$$-vector $$x$$ into a real number $$y$$. Then, seeing this functional as a matrix $$M$$, and $$x$$ as an $$n\times 1$$ matrix, and $$y$$ a $$1\times 1$$ matrix (trivially, a real number) respectively, if $$Mx=y$$ then, by dimension reasons, $$M$$ must be a $$1\times n$$ matrix; that is, $$M$$ must be a row vector.

If $$V$$ consists of the space of geometrical vectors in the plane, then the level curves of an element of $$V^*$$ form a family of parallel lines in $$V$$, because the range is 1-dimensional, so that every point in the range is a multiple of any one nonzero element. So an element of $$V^*$$ can be intuitively thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, it suffices to determine which of the lines the vector lies on. Informally, this "counts" how many lines the vector crosses. More generally, if $$V$$ is a vector space of any dimension, then the level sets of a linear functional in $$V^*$$  are parallel hyperplanes in $$V$$, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes.

Infinite-dimensional case
If $$V$$ is not finite-dimensional but has a basis $$\mathbf{e}_\alpha$$ indexed by an infinite set $$A$$, then the same construction as in the finite-dimensional case yields linearly independent elements $$\mathbf{e}^\alpha$$ ($$\alpha\in A$$) of the dual space, but they will not form a basis.

For instance, consider the space $$\R^\infty$$, whose elements are those sequences of real numbers that contain only finitely many non-zero entries, which has a basis indexed by the natural numbers $$\N$$. For $$i \in \N$$, $$\mathbf{e}_i$$ is the sequence consisting of all zeroes except in the $$i$$-th position, which is 1. The dual space of $$\R^\infty$$ is (isomorphic to) $$\R^\N$$, the space of all sequences of real numbers: each real sequence $$(a_n)$$ defines a function where the element $$(x_n)$$ of $$\R^\infty$$ is sent to the number

 * $$\sum_n a_nx_n,$$

which is a finite sum because there are only finitely many nonzero $$x_n$$. The dimension of $$\R^\infty$$ is countably infinite, whereas $$\R^\N$$ does not have a countable basis.

This observation generalizes to any infinite-dimensional vector space $$V$$ over any field $$F$$: a choice of basis $$\{\mathbf{e}_\alpha:\alpha\in A\}$$ identifies $$V$$ with the space $$(F^A)_0$$ of functions $$f:A\to F$$ such that $$f_\alpha=f(\alpha)$$ is nonzero for only finitely many $$\alpha\in A$$, where such a function $$f$$ is identified with the vector

 * $$\sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha$$

in $$V$$ (the sum is finite by the assumption on $$f$$, and any $$v\in V$$ may be written uniquely in this way by the definition of the basis).

The dual space of $$V$$ may then be identified with the space $$F^A$$ of all functions from $$A$$ to $$F$$: a linear functional $$T$$ on $$V$$ is uniquely determined by the values $$\theta_\alpha=T(\mathbf{e}_\alpha)$$ it takes on the basis of $$V$$, and any function $$\theta:A\to F$$ (with $$\theta(\alpha)=\theta_\alpha$$) defines a linear functional $$T$$ on $$V$$ by

 * $$T\left (\sum_{\alpha\in A} f_\alpha \mathbf{e}_\alpha\right) = \sum_{\alpha \in A} f_\alpha T(e_\alpha) = \sum_{\alpha\in A} f_\alpha \theta_\alpha.$$

Again, the sum is finite because $$f_\alpha$$ is nonzero for only finitely many $$\alpha$$.

The set $$(F^A)_0$$ may be identified (essentially by definition) with the direct sum of infinitely many copies of $$F$$ (viewed as a 1-dimensional vector space over itself) indexed by $$A$$, i.e. there are linear isomorphisms
 * $$ V\cong (F^A)_0\cong\bigoplus_{\alpha\in A} F.$$

On the other hand, $$F^A$$ is (again by definition), the direct product of infinitely many copies of $$F$$ indexed by $$A$$, and so the identification
 * $$V^* \cong \left (\bigoplus_{\alpha\in A}F\right )^* \cong \prod_{\alpha\in A}F^* \cong \prod_{\alpha\in A}F \cong F^A$$

is a special case of a general result relating direct sums (of modules) to direct products.

If a vector space is not finite-dimensional, then its (algebraic) dual space is always of larger dimension (as a cardinal number) than the original vector space. This is in contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional.

The proof of this inequality between dimensions results from the following.

If $$V$$ is an infinite-dimensional $$F$$-vector space, the arithmetical properties of cardinal numbers implies that
 * $$\mathrm{dim}(V)=|A|<|F|^{|A|}=|V^\ast|=\mathrm{max}(|\mathrm{dim}(V^\ast)|, |F|),$$

where cardinalities are denoted as absolute values. For proving that $$\mathrm{dim}(V)< \mathrm{dim}(V^*),$$ it suffices to prove that $$|F|\le |\mathrm{dim}(V^\ast)|,$$ which can be done with an argument similar to Cantor's diagonal argument. The exact dimension of the dual is given by the Erdős–Kaplansky theorem.

Bilinear products and dual spaces
If V is finite-dimensional, then V is isomorphic to V∗. But there is in general no natural isomorphism between these two spaces. Any bilinear form $⟨·,·⟩$ on V gives a mapping of V into its dual space via

 * $$v\mapsto \langle v, \cdot\rangle$$

where the right hand side is defined as the functional on V taking each $V × V$ to $⟨·,·⟩$. In other words, the bilinear form determines a linear mapping

 * $$\Phi_{\langle\cdot,\cdot\rangle} : V\to V^*$$

defined by

 * $$\left[\Phi_{\langle\cdot,\cdot\rangle}(v), w\right] = \langle v, w\rangle.$$

If the bilinear form is nondegenerate, then this is an isomorphism onto a subspace of V∗. If V is finite-dimensional, then this is an isomorphism onto all of V∗. Conversely, any isomorphism $$\Phi$$ from V to a subspace of V∗ (resp., all of V∗ if V is finite dimensional) defines a unique nondegenerate bilinear form $w ∈ V$ on V by

 * $$ \langle v, w \rangle_\Phi = (\Phi (v))(w) = [\Phi (v), w].\,$$

Thus there is a one-to-one correspondence between isomorphisms of V to a subspace of (resp., all of) V∗ and nondegenerate bilinear forms on V.

If the vector space V is over the complex field, then sometimes it is more natural to consider sesquilinear forms instead of bilinear forms. In that case, a given sesquilinear form $⟨v, w⟩$ determines an isomorphism of V with the complex conjugate of the dual space

\Phi_{\langle \cdot, \cdot \rangle} : V\to \overline{V^*}. $$ The conjugate of the dual space $$\overline{V^*}$$ can be identified with the set of all additive complex-valued functionals $$ \langle \cdot, \cdot \rangle_{\Phi} $$ such that

f(\alpha v) = \overline{\alpha}f(v). $$

Injection into the double-dual
There is a natural homomorphism $$\Psi$$ from $$V$$ into the double dual $$V^{**}=\{\Phi:V^*\to F:\Phi\ \mathrm{linear}\}$$, defined by $$(\Psi(v))(\varphi)=\varphi(v)$$ for all $$v\in V, \varphi\in V^*$$. In other words, if $$\mathrm{ev}_v:V^*\to F$$ is the evaluation map defined by $$\varphi \mapsto \varphi(v)$$, then $$\Psi: V \to V^{**}$$ is defined as the map $$v\mapsto\mathrm{ev}_v$$. This map $$\Psi$$ is always injective; and it is always an isomorphism if $$V$$ is finite-dimensional. Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.

Transpose of a linear map
If $⟨·,·⟩$ is a linear map, then the transpose (or dual) $f : V → C$ is defined by

f^*(\varphi) = \varphi \circ f \, $$ for every $$\varphi \in W^*$$. The resulting functional $$f^* (\varphi)$$ in $$V^*$$ is called the pullback of $$\varphi$$ along $$f$$.

The following identity holds for all $$\varphi \in W^*$$ and $$v \in V$$:

[f^*(\varphi),\, v] = [\varphi,\, f(v)], $$ where the bracket [·,·] on the left is the natural pairing of V with its dual space, and that on the right is the natural pairing of W with its dual. This identity characterizes the transpose, and is formally similar to the definition of the adjoint.

The assignment $f : V → W$ produces an injective linear map between the space of linear operators from V to W and the space of linear operators from W to V; this homomorphism is an isomorphism if and only if W is finite-dimensional. If $f : W → V$ then the space of linear maps is actually an algebra under composition of maps, and the assignment is then an antihomomorphism of algebras, meaning that $f ↦ f$. In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself. It is possible to identify (f)$∗$ with f using the natural injection into the double dual.

If the linear map f is represented by the matrix A with respect to two bases of V and W, then f is represented by the transpose matrix AT with respect to the dual bases of W and V, hence the name. Alternatively, as f is represented by A acting on the left on column vectors, f is represented by the same matrix acting on the right on row vectors. These points of view are related by the canonical inner product on Rn, which identifies the space of column vectors with the dual space of row vectors.

Quotient spaces and annihilators
Let $$S$$ be a subset of $$V$$. The annihilator of $$S$$ in $$V^*$$, denoted here $$S^0$$, is the collection of linear functionals $$f\in V^*$$ such that $$[f,s]=0$$ for all $$s\in S$$. That is, $$S^0$$ consists of all linear functionals $$f:V\to F$$ such that the restriction to $$S$$ vanishes: $$f|_S = 0$$. Within finite dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement.

The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: $$\{ 0 \}^0 = V^*$$, and the annihilator of the whole space is just the zero covector: $$V^0 = \{ 0 \} \subseteq V^*$$. Furthermore, the assignment of an annihilator to a subset of $$V$$ reverses inclusions, so that if $$\{ 0 \} \subseteq S\subseteq T\subseteq V$$, then

\{ 0 \} \subseteq T^0 \subseteq S^0 \subseteq V^*. $$

If $$A$$ and $$B$$ are two subsets of $$V$$ then

A^0 + B^0 \subseteq (A \cap B)^0. $$ If $$(A_i)_{i\in I}$$ is any family of subsets of $$V$$ indexed by $$i$$ belonging to some index set $$I$$, then

\left( \bigcup_{i\in I} A_i \right)^0 = \bigcap_{i\in I} A_i^0. $$ In particular if $$A$$ and $$B$$ are subspaces of $$V$$ then

(A + B)^0 = A^0 \cap B^0 $$ and

(A \cap B)^0 = A^0 + B^0. $$

If $$V$$ is finite-dimensional and $$W$$ is a vector subspace, then

W^{00} = W $$ after identifying $$W$$ with its image in the second dual space under the double duality isomorphism $$V\approx V^{**}$$. In particular, forming the annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector space.

If $$W$$ is a subspace of $$V$$ then the quotient space $$V/W$$ is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional $$f:V\to F$$ factors through $$V/W$$ if and only if $$W$$ is in the kernel of $$f$$. There is thus an isomorphism
 * $$ (V/W)^* \cong W^0 .$$

As a particular consequence, if $$V$$ is a direct sum of two subspaces $$A$$ and $$B$$, then $$V^*$$ is a direct sum of $$A^0$$ and $$B^0$$.

Dimensional analysis
The dual space is analogous to a "negative"-dimensional space. Most simply, since a vector $$v \in V$$ can be paired with a covector $$\varphi \in V^*$$ by the natural pairing $$\langle x, \varphi \rangle := \varphi (x) \in F$$ to obtain a scalar, a covector can "cancel" the dimension of a vector, similar to reducing a fraction. Thus while the direct sum $$V \oplus V^*$$ is a $2n$-dimensional space (if $V$ is $n$-dimensional), $V^*$ behaves as an $(-n)$-dimensional space, in the sense that its dimensions can be canceled against the dimensions of $V$. This is formalized by tensor contraction.

This arises in physics via dimensional analysis, where the dual space has inverse units. Under the natural pairing, these units cancel, and the resulting scalar value is dimensionless, as expected. For example, in (continuous) Fourier analysis, or more broadly time–frequency analysis: given a one-dimensional vector space with a unit of time $t$, the dual space has units of frequency: occurrences per unit of time (units of $1/t$). For example, if time is measured in seconds, the corresponding dual unit is the inverse second: over the course of 3 seconds, an event that occurs 2 times per second occurs a total of 6 times, corresponding to $$3s \cdot 2s^{-1} = 6$$. Similarly, if the primal space measures length, the dual space measures inverse length.

Continuous dual space
When dealing with topological vector spaces, the continuous linear functionals from the space into the base field $$\mathbb{F} = \Complex$$ (or $$\R$$) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space $$V^*$$, denoted by $$V'$$. For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space".

For a topological vector space $$V$$ its continuous dual space, or topological dual space, or just dual space  (in the sense of the theory of topological vector spaces) $$V'$$ is defined as the space of all continuous linear functionals $$\varphi:V\to{\mathbb F}$$.

Important examples for continuous dual spaces are the space of compactly supported test functions $$\mathcal{D}$$ and its dual $$\mathcal{D}',$$ the space of arbitrary distributions (generalized functions); the space of arbitrary test functions $$\mathcal{E}$$ and its dual $$\mathcal{E}',$$ the space of compactly supported distributions; and the space of rapidly decreasing test functions $$\mathcal{S},$$ the Schwartz space, and its dual $$\mathcal{S}',$$ the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.

Properties
If $X$ is a Hausdorff topological vector space (TVS), then the continuous dual space of $X$ is identical to the continuous dual space of the completion of $X$.

Topologies on the dual
There is a standard construction for introducing a topology on the continuous dual $$V'$$ of a topological vector space $$V$$. Fix a collection $$\mathcal{A}$$ of bounded subsets of $$V$$. This gives the topology on $$V$$ of uniform convergence on sets from $$\mathcal{A},$$ or what is the same thing, the topology generated by seminorms of the form

 * $$\|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,$$

where $$\varphi$$ is a continuous linear functional on $$V$$, and $$A$$ runs over the class $$\mathcal{A}.$$

This means that a net of functionals $$\varphi_i$$ tends to a functional $$\varphi$$ in $$V'$$ if and only if

 * $$\text{ for all } A\in\mathcal{A}\qquad \|\varphi_i-\varphi\|_A = \sup_{x\in A} |\varphi_i(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow} 0. $$

Usually (but not necessarily) the class $$\mathcal{A}$$ is supposed to satisfy the following conditions:
 * Each point $$x$$ of $$V$$ belongs to some set $$A\in\mathcal{A}$$:
 * $$\text{ for all } x \in V\quad \text{ there exists some } A \in \mathcal{A}\quad \text{ such that } x \in A.$$
 * Each two sets $$A \in \mathcal{A}$$ and $$B \in \mathcal{A}$$ are contained in some set $$C \in \mathcal{A}$$:
 * $$\text{ for all } A, B \in \mathcal{A}\quad \text{ there exists some } C \in \mathcal{A}\quad \text{ such that } A \cup B \subseteq C.$$
 * $$\mathcal{A}$$ is closed under the operation of multiplication by scalars:
 * $$\text{ for all } A \in \mathcal{A}\quad \text{ and all } \lambda \in {\mathbb F}\quad \text{ such that } \lambda \cdot A \in \mathcal{A}.$$

If these requirements are fulfilled then the corresponding topology on $$V'$$ is Hausdorff and the sets

 * $$U_A ~=~ \left \{ \varphi \in V' ~:~ \quad \|\varphi\|_A < 1 \right \},\qquad \text{ for } A \in \mathcal{A}$$

form its local base.

Here are the three most important special cases. If $$V$$ is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on $$V'$$ is normed (in fact a Banach space if the field of scalars is complete), with the norm
 * The strong topology on $$V'$$ is the topology of uniform convergence on bounded subsets in $$V$$ (so here $$\mathcal{A}$$ can be chosen as the class of all bounded subsets in $$V$$).
 * $$\|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.$$

 * The stereotype topology on $$V'$$ is the topology of uniform convergence on totally bounded sets in $$V$$ (so here $$\mathcal{A}$$ can be chosen as the class of all totally bounded subsets in $$V$$).
 * The weak topology on $$V'$$ is the topology of uniform convergence on finite subsets in $$V$$ (so here $$\mathcal{A}$$ can be chosen as the class of all finite subsets in $$V$$).

Each of these three choices of topology on $$V'$$ leads to a variant of reflexivity property for topological vector spaces:
 * If $$V'$$ is endowed with the strong topology, then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called reflexive.
 * If $$V'$$ is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of stereotype spaces: the spaces reflexive in this sense are called stereotype.
 * If $$V'$$ is endowed with the weak topology, then the corresponding reflexivity is presented in the theory of dual pairs: the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.

Examples
Let 1 < p < ∞ be a real number and consider the Banach space ℓ p of all sequences $V = W$ for which

 * $$\|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{\frac{1}{p}} < \infty.$$

Define the number q by $(fg)∗ = gf$. Then the continuous dual of ℓ p is naturally identified with ℓ q: given an element $$\varphi \in (\ell^p)'$$, the corresponding element of $a = (a_{n})$ is the sequence $$(\varphi(\mathbf {e}_n))$$ where $$\mathbf {e}_n$$ denotes the sequence whose $n$-th term is 1 and all others are zero. Conversely, given an element $1/p + 1/q = 1$, the corresponding continuous linear functional $$\varphi$$ on $ℓ^{ q}$ is defined by

 * $$\varphi (\mathbf{b}) = \sum_n a_n b_n$$

for all $a = (a_{n}) ∈ ℓ^{ q}$ (see Hölder's inequality).

In a similar manner, the continuous dual of $ℓ^{ p}$ is naturally identified with $b = (b_{n}) ∈ ℓ^{ p}$ (the space of bounded sequences). Furthermore, the continuous duals of the Banach spaces c (consisting of all convergent sequences, with the supremum norm) and c0 (the sequences converging to zero) are both naturally identified with $ℓ^{ 1}$.

By the Riesz representation theorem, the continuous dual of a Hilbert space is again a Hilbert space which is anti-isomorphic to the original space. This gives rise to the bra–ket notation used by physicists in the mathematical formulation of quantum mechanics.

By the Riesz–Markov–Kakutani representation theorem, the continuous dual of certain spaces of continuous functions can be described using measures.

Transpose of a continuous linear map
If $ℓ^{ ∞}$ is a continuous linear map between two topological vector spaces, then the (continuous) transpose $ℓ^{ 1}$ is defined by the same formula as before:

 * $$T'(\varphi) = \varphi \circ T, \quad \varphi \in W'.$$

The resulting functional $T : V → W$ is in $T′ : W′ → V′$. The assignment $T′(φ)$ produces a linear map between the space of continuous linear maps from V to W and the space of linear maps from $V′$ to $T → T′$. When T and U are composable continuous linear maps, then

 * $$(U \circ T)' = T' \circ U'.$$

When V and W are normed spaces, the norm of the transpose in$W′$ is equal to that of T in $V′$. Several properties of transposition depend upon the Hahn–Banach theorem. For example, the bounded linear map T has dense range if and only if the transpose $L(W′, V′)$ is injective.

When T is a compact linear map between two Banach spaces V and W, then the transpose $L(V, W)$ is compact. This can be proved using the Arzelà–Ascoli theorem.

When V is a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual $T′$. For every bounded linear map T on V, the transpose and the adjoint operators are linked by

 * $$i_V \circ T^* = T' \circ i_V.$$

When T is a continuous linear map between two topological vector spaces V and W, then the transpose $T′$ is continuous when $V′$ and $T′$ are equipped with "compatible" topologies: for example, when for $W′$ and $V′$, both duals $X = V$ have the strong topology $X = W$ of uniform convergence on bounded sets of X, or both have the weak-∗ topology $X′$ of pointwise convergence on X. The transpose $β(X′, X)$ is continuous from $σ(X′, X)$ to $T′$, or from $β(W′, W)$ to $β(V′, V)$.

Annihilators
Assume that W is a closed linear subspace of a normed space V, and consider the annihilator of W in $σ(W′, W)$,

 * $$W^\perp = \{ \varphi \in V' : W \subseteq \ker \varphi\}.$$

Then, the dual of the quotient $σ(V′, V)$ can be identified with W⊥, and the dual of W can be identified with the quotient $V′$. Indeed, let P denote the canonical surjection from V onto the quotient $V / W $; then, the transpose $V′ / W^{⊥}$ is an isometric isomorphism from $V / W $ into $P′$, with range equal to W⊥. If j denotes the injection map from W into V, then the kernel of the transpose $(V / W )′$ is the annihilator of W:
 * $$\ker (j') = W^\perp$$

and it follows from the Hahn–Banach theorem that $V′$ induces an isometric isomorphism $j′$.

Further properties
If the dual of a normed space $V$ is separable, then so is the space $V$ itself. The converse is not true: for example, the space $j′$ is separable, but its dual $V′ / W^{⊥} → W′$ is not.

Double dual
In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator $ℓ^{ 1}$ from a normed space V into its continuous double dual $ℓ^{ ∞}$, defined by

 * $$ \Psi(x)(\varphi) = \varphi(x), \quad x \in V, \ \varphi \in V' .$$

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning $⟨x_{1}, x_{2}⟩$ for all $x_{1} + x_{2}$. Normed spaces for which the map Ψ is a bijection are called reflexive.

When V is a topological vector space then Ψ(x) can still be defined by the same formula, for every $Ψ : V → V′′$, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to { 0 } and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual $V′′$ of the continuous dual, again as a consequence of the Hahn–Banach theorem.

Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual $‖ Ψ(x) ‖ = ‖ x ‖$, so that the continuous double dual $x ∈ V$ is not uniquely defined as a set. Saying that Ψ maps from V to $x ∈ V$, or in other words, that Ψ(x) is continuous on $V′^{∗}$ for every $V′$, is a reasonable minimal requirement on the topology of $V′′$, namely that the evaluation mappings

 * $$ \varphi \in V' \mapsto \varphi(x), \quad x \in V, $$

be continuous for the chosen topology on $V′′$. Further, there is still a choice of a topology on $V′$, and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.

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

title: Dual space


text:
Dianetics

Dianetics is a set of ideas and practices invented in 1950 by science fiction writer L. Ron Hubbard regarding the human mind. Dianetics was originally conceived as a form of psychological treatment, but was rejected by the psychological and medical establishments as pseudoscientific. Dianetics was the precursor to Scientology and has since been incorporated into it.

Premise
The word Dianetics was coined from Greek dia meaning "through" and nous meaning "mind".

Dianetics theory describes the human mind as two parts: the conscious "analytical mind" and the subconscious "reactive mind". The purpose of Dianetics technique, called "auditing", is to erase the contents of the reactive mind—the holder of painful and destructive emotions which can act on a person as posthypnotic suggestions. In auditing, the person is asked questions intended to help them locate and deal with painful past experiences.

Dianetics theory posits that "the basic principle of existence is to survive" and that the basic personality of humans is sincere, intelligent, and good. The drive for goodness and survival is distorted and inhibited by aberrations (deviations from rational thinking). Hubbard claimed that Dianetics could increase intelligence, eliminate unwanted emotions and alleviate a wide range of illnesses he believed to be psychosomatic. Conditions purportedly treatable with Dianetics included arthritis, allergies, asthma, some coronary difficulties, eye trouble, ulcers, migraine headaches, and sexual deviation.

History
According to Hubbard, when he was sedated for a dental operation in 1938, he had a near-death experience which inspired him to write the manuscript Excalibur. Though it was never published, this work would allegedly become the basis for Dianetics. The first publication on Dianetics was Dianetics: The Evolution of a Science, an article by Hubbard in Astounding Science Fiction (cover date May 1950). This was followed by the book Dianetics: The Modern Science of Mental Health (DMSMH) published May 9, 1950. In these works Hubbard claimed that the source of all psychological pain, and therefore the cause of mental and physical health problems, was a form of memory known as "engrams". According to Hubbard, individuals could reach a state he named "Clear" when all of their engrams had been removed through talking with an "auditor".

While the technique was not accepted by the medical and scientific establishment, in the first two years of its publication DMSMH sold over 100,000 copies. Publication of DMSMH brought in a flood of revenue, which Hubbard used to establish Dianetics foundations in six major American cities. Two of the strongest initial supporters of Dianetics in the 1950s were John W. Campbell, editor of Astounding Science Fiction, and Joseph Augustus Winter, a writer and medical physician. Campbell published some of Hubbard's short stories, and Winter hoped that his own colleagues would likewise be attracted to Hubbard's Dianetics system.

Readers formed groups to study and practice Dianetics technique. According to sociologist Roy Wallis, this period was one of "excited experimentation" and Hubbard's work was regarded as "an initial exploration to be developed further by others". Per Wallis, it was Dianetics' popularity as a lay psychotherapy that contributed to the Dianetics Foundation's downfall. Most people read the book, tried it out, then put it down. The remaining practitioners had no ties to the Foundation. Factions formed and followers challenged Hubbard's movement and his authority. The craze of 1950–51 was dead by 1952.

In 1951, with debts piled up and facing bankruptcy, Don Purcell, a wealthy Dianetics follower from Wichita, bailed out the foundation. It was short-lived, however, and the foundation fell to bankruptcy in 1952. Hubbard fled to Phoenix, Arizona, having lost the foundation, the rights to Dianetics, and the DMSMH copyrights to Purcell. Hubbard sued and in 1954 Purcell settled by giving the copyrights back to Hubbard.

In Phoenix, Hubbard created "Scientology"; its techniques were intended to rehabilitate a person so that they might reach their full potential as a spiritual being. Dianetics was incorporated into Scientology. In 1978, Hubbard introduced "New Era Dianetics" (NED) and New Era Dianetics for OTs, and added them to The Bridge to Total Freedom.

Concepts
In the book, Dianetics: The Modern Science of Mental Health, Hubbard describes techniques that he suggests can rid individuals of fears and psychosomatic illnesses. A basic idea in Dianetics is that the mind consists of two parts: the "analytical mind" and the "reactive mind". The "reactive mind", the mind which operates when a person is physically unconscious, acts as a record of shock, trauma, pain, and otherwise harmful memories. Experiences such as these, stored in the "reactive mind" are dubbed "engrams". Dianetics is proposed as a method to erase these engrams in the reactive mind to achieve a state of clear.

In Dianetics, the unconscious or reactive mind is described as a collection of "mental image pictures", which contain the recorded experience of past moments of unconsciousness, including all sensory perceptions and feelings involved, ranging from pre-natal experiences, infancy and childhood, to even the traumatic feelings associated with events from past lives and extraterrestrial cultures. The type of mental image picture created during a period of unconsciousness involves the exact recording of a painful experience. Hubbard called this phenomenon an engram, and defined it as "a complete recording of a moment of unconsciousness containing physical pain or painful emotion and all perceptions."

Hubbard proposed that these engrams caused "aberrations" (deviations from rational thinking) in the mind, which produced lasting adverse physical and emotional effects. When the analytical (conscious) mind shut down during these moments, events and perceptions of this period were stored as engrams in the unconscious or reactive mind. In Hubbard's earliest publications on the subject, engrams were variously referred to as norn, impediment, and comanome before "engram" was adapted from its existing usage at the suggestion of Joseph Augustus Winter, MD. Some commentators noted Dianetics's blend of science fiction and occult orientations at the time.

Hubbard claimed that these engrams were the cause of almost all psychological and physical problems. In addition to physical pain, engrams could include words or phrases spoken in the vicinity while the patient was unconscious. For instance, Winter cites the example of a patient with a persistent headache supposedly tracing the problem to a doctor saying, "Take him now", during the patient's birth.

"[The reactive mind] can give a man arthritis, bursitis, asthma, allergies, sinusitis, coronary trouble, high blood pressure ... And it is the only thing in the human being which can produce these effects ... Discharge the content of [the reactive mind] and the arthritis vanishes, myopia gets better, heart illness decreases, asthma disappears, stomachs function properly and the whole catalog of ills goes away and stays away."

According to Bent Corydon, Hubbard created the illusion that Dianetics was the first psychotherapy to address traumatic experiences in their own time, but others had done so before as standard procedure. Hugh Urban wrote it was clear that Hubbard's work had been influenced by Carl Jung, Sigmund Freud and Otto Rank, and Hubbard himself mentioned similarities between Dianetics and Freud.

Hubbard claimed that by using Dianetics technique the reactive mind could be emptied of all engrams; "cleared" of its contents. A person who has completed this process would be "Clear". The benefits of Clear might include a higher IQ, better relationships, or career success.

Procedure
The procedure of Dianetics therapy (known as auditing) is a two-person activity. One person, the "auditor", guides the other person, the preclear, through the procedures. The preclear's job is to look at their mind and talk to the auditor. The auditor acknowledges what the preclear says and controls the process.

The auditor and preclear sit down facing each other. After getting settled, the auditor tells the preclear to close their eyes and locate something that happened to them in the past. The preclear tells the auditor what happened in the incident like he is re-experiencing it again. The auditor coaxes the preclear to recall as much as possible, and goes back over the incident several times until the preclear is cheerful about it, at which point the auditor may end the session or find another incident and repeat the process.

Therapeutic claims
"The slick craftsman of mass-production science-fiction, mustering his talents and energies for a supreme effort, produces [...] a fictional science. Had dianetics been presented as fiction [...] it might have been, like other ingenious science-fiction, good entertainment."

In August 1950, amidst the success of Dianetics, Hubbard held a demonstration in Los Angeles' Shrine Auditorium where he presented a young woman called Sonya Bianca (a pseudonym) to a large audience including many reporters and photographers as "the world's first Clear". Despite Hubbard's claim that she had "full and perfect recall of every moment of her life", Bianca proved unable to answer questions from the audience testing her memory and analytical abilities, including the question of the color of Hubbard's tie. Hubbard explained Bianca's failure to display her promised powers of recall to the audience by saying that he had used the word "now" in calling her to the stage, and thus inadvertently froze her in "present time", which blocked her abilities. Later, in the late 1950s, Hubbard would claim that several people had reached the state of Clear by the time he presented Bianca as the world's first; these others, Hubbard said, he had successfully cleared in the late 1940s while working incognito in Hollywood posing as a swami. In 1966, Hubbard declared South African Scientologist John McMaster to be the first true Clear.

Hubbard claimed, in an interview with The New York Times in November 1950, that "he had already submitted proof of claims made in the book to a number of scientists and associations." He added that the public as well as proper organizations were entitled to such proof and that he was ready and willing to give such proof in detail. In January 1951, the Hubbard Dianetic Research Foundation of Elizabeth, New Jersey, published Dianetic Processing: A Brief Survey of Research Projects and Preliminary Results, a booklet providing the results of psychometric tests conducted on 88 people undergoing Dianetics therapy. It presents case histories and a number of X-ray plates to support claims that Dianetics had cured "aberrations" including manic depression, asthma, arthritis, colitis and "overt homosexuality", and that after Dianetic processing, test subjects experienced significantly increased scores on a standardized IQ test. The report's subjects are not identified by name, but one of them is clearly Hubbard himself ("Case 1080A, R. L.").

The authors provide no qualifications, although they are described in Hubbard's book Science of Survival (where some results of the same study were reprinted) as psychotherapists. Critics of Dianetics are skeptical of this study, both because of the bias of the source and because the researchers appear to ascribe all physical benefits to Dianetics without considering possible outside factors; in other words, the report lacks any scientific controls. Winter was originally an associate of Hubbard and an early adopter of Dianetics, but by the end of 1950 had cut ties with Hubbard and written an account of his personal experiences with Dianetics. He described Hubbard as "absolutistic and authoritarian", and criticized the Hubbard Dianetic Research Foundation for failing to undertake "precise scientific research into the functioning of the mind". He also recommended that auditing be done by experts only and that it was dangerous for laymen to audit each other. Hubbard writes: "Again, Dianetics is not being released to a profession, for no profession could encompass it."

Scientific rejection
Hubbard's original book on Dianetics attracted highly critical reviews from science and medical writers and organizations. The American Psychological Association passed a resolution in 1950 calling "attention to the fact that these claims are not supported by empirical evidence of the sort required for the establishment of scientific generalizations." Subsequently, Dianetics has achieved no acceptance as a scientific theory, and scientists cite Dianetics as an example of a pseudoscience.

Few scientific investigations into the effectiveness of Dianetics have been published. Professor John A. Lee states in his 1970 evaluation of Dianetics:

"Objective experimental verification of Hubbard's physiological and psychological doctrines is lacking. To date, no regular scientific agency has established the validity of his theories of prenatal perception and engrams, or cellular memory, or Dianetic reverie, or the effects of Scientology auditing routines. Existing knowledge contradicts Hubbard's theory of recording of perceptions during periods of unconsciousness."

The MEDLINE database records two independent scientific studies on Dianetics, both conducted in the 1950s under the auspices of New York University. Harvey Jay Fischer tested Dianetic therapy against three claims made by proponents and found it does not effect any significant changes in intellectual functioning, mathematical ability, or the degree of personality conflicts; Jack Fox tested Hubbard's thesis regarding recall of engrams, with the assistance of the Dianetic Research Foundation, and could not substantiate it.

Commentators from a variety of backgrounds have described Dianetics as an example of pseudoscience. For example, philosophy professor Robert Carroll points to Dianetics' lack of empirical evidence:

"What Hubbard touts as a science of mind lacks one key element that is expected of a science: empirical testing of claims. The key elements of Hubbard's so-called science don't seem testable, yet he repeatedly claims that he is asserting only scientific facts and data from many experiments. It isn't even clear what such 'data' would look like. Most of his data is in the form of anecdotes and speculations ... Such speculation is appropriate in fiction, but not in science."

The validity and practice of auditing have been questioned by a variety of non-Scientologist commentators. Commenting on the example cited by Winter, the science writer Martin Gardner asserts that "nothing could be clearer from the above dialogue than the fact that the dianetic explanation for the headache existed only in the mind of the therapist, and that it was with considerable difficulty that the patient was maneuvered into accepting it."

Other critics and medical experts have suggested that Dianetic auditing is a form of hypnosis. Hubbard, who had previously used hypnosis for entertainment purposes, strongly denied this connection and cautioned against hypnosis in Dianetics auditing. Professor Richard J. Ofshe, a leading expert on false memories, suggests that the feeling of well-being reported by preclear at the end of an auditing session may be induced by post-hypnotic suggestion. Other researchers have identified quotations in Hubbard's work suggesting evidence that false memories were created in Dianetics, specifically in the form of birth and pre-birth memories.

According to an article by physician Martin Gumpert, "Hubbard's concept of psychosomatic disease is definitely wrong. Psychosomatic ailments are not simply caused by emotional disturbances: they are diseases in which the emotional and the organic factor are closely involved and interdependent."

"But even the limited good that dianetics may do by introducing a single, narrowly-defined role-playing technique into interpersonal relations is probably more than offset by the damage it can do with its accompanying pretentious and nonsensical doctrines. [...T]hose who are helped by dianetics will necessarily be kept at a low level of intellectual and emotional maturity by the nonsense they have absorbed in order to be helped. The lure of the pseudoscientific vocabulary and promises of dianetics cannot but condemn thousands who are beginning to emerge from scientific illiteracy to a continuation of their susceptibility to word-magic and semantic hash. —S. I. Hayakawa"

title: Dianetics


text:
Data warehouse

In computing, a data warehouse (DW or DWH), also known as an enterprise data warehouse (EDW), is a system used for reporting and data analysis and is considered a core component of business intelligence. Data warehouses are central repositories of integrated data from one or more disparate sources. They store current and historical data in one single place that are used for creating reports. This is beneficial for companies as it enables them to interrogate and draw insights from their data and make decisions.

The data stored in the warehouse is uploaded from the operational systems (such as marketing or sales). The data may pass through an operational data store and may require data cleansing for additional operations to ensure data quality before it is used in the data warehouse for reporting.

Extract, transform, load (ETL) and extract, load, transform (ELT) are the two main approaches used to build a data warehouse system.

ETL-based data warehousing
The typical extract, transform, load (ETL)-based data warehouse uses staging, data integration, and access layers to house its key functions. The staging layer or staging database stores raw data extracted from each of the disparate source data systems. The integration layer integrates disparate data sets by transforming the data from the staging layer, often storing this transformed data in an operational data store (ODS) database. The integrated data are then moved to yet another database, often called the data warehouse database, where the data is arranged into hierarchical groups, often called dimensions, and into facts and aggregate facts. The combination of facts and dimensions is sometimes called a star schema. The access layer helps users retrieve data.

The main source of the data is cleansed, transformed, catalogued, and made available for use by managers and other business professionals for data mining, online analytical processing, market research and decision support. However, the means to retrieve and analyze data, to extract, transform, and load data, and to manage the data dictionary are also considered essential components of a data warehousing system. Many references to data warehousing use this broader context. Thus, an expanded definition of data warehousing includes business intelligence tools, tools to extract, transform, and load data into the repository, and tools to manage and retrieve metadata.

ELT-based data warehousing

ELT-based data warehousing gets rid of a separate ETL tool for data transformation. Instead, it maintains a staging area inside the data warehouse itself. In this approach, data gets extracted from heterogeneous source systems and are then directly loaded into the data warehouse, before any transformation occurs. All necessary transformations are then handled inside the data warehouse itself. Finally, the manipulated data gets loaded into target tables in the same data warehouse.

Benefits
A data warehouse maintains a copy of information from the source transaction systems. This architectural complexity provides the opportunity to:
 * Integrate data from multiple sources into a single database and data model. More congregation of data to single database so a single query engine can be used to present data in an ODS.
 * Mitigate the problem of database isolation level lock contention in transaction processing systems caused by attempts to run large, long-running analysis queries in transaction processing databases.
 * Maintain data history, even if the source transaction systems do not.
 * Integrate data from multiple source systems, enabling a central view across the enterprise. This benefit is always valuable, but particularly so when the organization has grown by merger.
 * Improve data quality, by providing consistent codes and descriptions, flagging or even fixing bad data.
 * Present the organization's information consistently.
 * Provide a single common data model for all data of interest regardless of the data's source.
 * Restructure the data so that it makes sense to the business users.
 * Restructure the data so that it delivers excellent query performance, even for complex analytic queries, without impacting the operational systems.
 * Add value to operational business applications, notably customer relationship management (CRM) systems.
 * Make decision–support queries easier to write.
 * Organize and disambiguate repetitive data.

Generic
The environment for data warehouses and marts includes the following:

 * Source systems that provide data to the warehouse or mart;
 * Data integration technology and processes that are needed to prepare the data for use;
 * Different architectures for storing data in an organization's data warehouse or data marts;
 * Different tools and applications for a variety of users;
 * Metadata, data quality, and governance processes must be in place to ensure that the warehouse or mart meets its purposes.

In regards to source systems listed above, R. Kelly Rainer states, "A common source for the data in data warehouses is the company's operational databases, which can be relational databases".

Regarding data integration, Rainer states, "It is necessary to extract data from source systems, transform them, and load them into a data mart or warehouse".

Rainer discusses storing data in an organization's data warehouse or data marts.

Metadata is data about data. "IT personnel need information about data sources; database, table, and column names; refresh schedules; and data usage measures".

Today, the most successful companies are those that can respond quickly and flexibly to market changes and opportunities. A key to this response is the effective and efficient use of data and information by analysts and managers. A "data warehouse" is a repository of historical data that is organized by the subject to support decision-makers in the organization. Once data is stored in a data mart or warehouse, it can be accessed.

Related systems
A data mart is a simple form of a data warehouse that is focused on a single subject (or functional area), hence they draw data from a limited number of sources such as sales, finance or marketing. Data marts are often built and controlled by a single department within an organization. The sources could be internal operational systems, a central data warehouse, or external data. Denormalization is the norm for data modeling techniques in this system. Given that data marts generally cover only a subset of the data contained in a data warehouse, they are often easier and faster to implement.

Types of data marts include dependent, independent, and hybrid data marts.

Online analytical processing (OLAP) is characterized by a relatively low volume of transactions. Queries are often very complex and involve aggregations. For OLAP systems, response time is an effective measure. OLAP applications are widely used by Data Mining techniques. OLAP databases store aggregated, historical data in multi-dimensional schemas (usually star schemas). OLAP systems typically have a data latency of a few hours, as opposed to data marts, where latency is expected to be closer to one day. The OLAP approach is used to analyze multidimensional data from multiple sources and perspectives. The three basic operations in OLAP are Roll-up (Consolidation), Drill-down, and Slicing & Dicing.

Online transaction processing (OLTP) is characterized by a large number of short on-line transactions (INSERT, UPDATE, DELETE). OLTP systems emphasize very fast query processing and maintaining data integrity in multi-access environments. For OLTP systems, effectiveness is measured by the number of transactions per second. OLTP databases contain detailed and current data. The schema used to store transactional databases is the entity model (usually 3NF). Normalization is the norm for data modeling techniques in this system.

Predictive analytics is about finding and quantifying hidden patterns in the data using complex mathematical models that can be used to predict future outcomes. Predictive analysis is different from OLAP in that OLAP focuses on historical data analysis and is reactive in nature, while predictive analysis focuses on the future. These systems are also used for customer relationship management (CRM).

History
The concept of data warehousing dates back to the late 1980s when IBM researchers Barry Devlin and Paul Murphy developed the "business data warehouse". In essence, the data warehousing concept was intended to provide an architectural model for the flow of data from operational systems to decision support environments. The concept attempted to address the various problems associated with this flow, mainly the high costs associated with it. In the absence of a data warehousing architecture, an enormous amount of redundancy was required to support multiple decision support environments. In larger corporations, it was typical for multiple decision support environments to operate independently. Though each environment served different users, they often required much of the same stored data. The process of gathering, cleaning and integrating data from various sources, usually from long-term existing operational systems (usually referred to as legacy systems), was typically in part replicated for each environment. Moreover, the operational systems were frequently reexamined as new decision support requirements emerged. Often new requirements necessitated gathering, cleaning and integrating new data from "data marts" that was tailored for ready access by users.

Additionally, with the publication of The IRM Imperative (Wiley & Sons, 1991) by James M. Kerr, the idea of managing and putting a dollar value on an organization's data resources and then reporting that value as an asset on a balance sheet became popular. In the book, Kerr described a way to populate subject-area databases from data derived from transaction-driven systems to create a storage area where summary data could be further leveraged to inform executive decision-making. This concept served to promote further thinking of how a data warehouse could be developed and managed in a practical way within any enterprise.

Key developments in early years of data warehousing:

 * 1960s – General Mills and Dartmouth College, in a joint research project, develop the terms dimensions and facts.
 * 1970s – ACNielsen and IRI provide dimensional data marts for retail sales.
 * 1970s – Bill Inmon begins to define and discuss the term Data Warehouse.
 * 1975 – Sperry Univac introduces MAPPER (MAintain, Prepare, and Produce Executive Reports), a database management and reporting system that includes the world's first 4GL. It is the first platform designed for building Information Centers (a forerunner of contemporary data warehouse technology).
 * 1983 – Teradata introduces the DBC/1012 database computer specifically designed for decision support.
 * 1984 – Metaphor Computer Systems, founded by David Liddle and Don Massaro, releases a hardware/software package and GUI for business users to create a database management and analytic system.
 * 1988 – Barry Devlin and Paul Murphy publish the article "An architecture for a business and information system" where they introduce the term "business data warehouse".
 * 1990 – Red Brick Systems, founded by Ralph Kimball, introduces Red Brick Warehouse, a database management system specifically for data warehousing.
 * 1991 - James M. Kerr authors The IRM Imperative, which suggests data resources could be reported as an asset on a balance sheet, furthering commercial interest in the establishment of data warehouses.
 * 1991 – Prism Solutions, founded by Bill Inmon, introduces Prism Warehouse Manager, software for developing a data warehouse.
 * 1992 – Bill Inmon publishes the book Building the Data Warehouse.
 * 1995 – The Data Warehousing Institute, a for-profit organization that promotes data warehousing, is founded.
 * 1996 – Ralph Kimball publishes the book The Data Warehouse Toolkit.
 * 1998 – Focal modeling is implemented as an ensemble (hybrid) data warehouse modeling approach, with Patrik Lager as one of the main drivers.
 * 2000 – Dan Linstedt releases in the public domain the Data vault modeling, conceived in 1990 as an alternative to Inmon and Kimball to provide long-term historical storage of data coming in from multiple operational systems, with emphasis on tracing, auditing and resilience to change of the source data model.
 * 2008 – Bill Inmon, along with Derek Strauss and Genia Neushloss, publishes "DW 2.0: The Architecture for the Next Generation of Data Warehousing", explaining his top-down approach to data warehousing and coining the term, data-warehousing 2.0.
 * 2008 – Anchor modeling was formalized in a paper presented at the International Conference on Conceptual Modeling, and won the best paper award
 * 2012 – Bill Inmon develops and makes public technology known as "textual disambiguation". Textual disambiguation applies context to raw text and reformats the raw text and context into a standard data base format. Once raw text is passed through textual disambiguation, it can easily and efficiently be accessed and analyzed by standard business intelligence technology. Textual disambiguation is accomplished through the execution of textual ETL. Textual disambiguation is useful wherever raw text is found, such as in documents, Hadoop, email, and so forth.
 * 2013 – Data vault 2.0 was released, having some minor changes to the modeling method, as well as integration with best practices from other methodologies, architectures and implementations including agile and CMMI principles

Facts
A fact is a value, or measurement, which represents a fact about the managed entity or system.

Facts, as reported by the reporting entity, are said to be at raw level; e.g., in a mobile telephone system, if a BTS (base transceiver station) receives 1,000 requests for traffic channel allocation, allocates for 820, and rejects the remaining, it would report three facts or measurements to a management system:

Facts at the raw level are further aggregated to higher levels in various dimensions to extract more service or business-relevant information from it. These are called aggregates or summaries or aggregated facts.

For instance, if there are three BTS in a city, then the facts above can be aggregated from the BTS to the city level in the network dimension. For example:

Dimensional versus normalized approach for storage of data
There are three or more leading approaches to storing data in a data warehouse – the most important approaches are the dimensional approach and the normalized approach.

The dimensional approach refers to Ralph Kimball's approach in which it is stated that the data warehouse should be modeled using a Dimensional Model/star schema. The normalized approach, also called the 3NF model (Third Normal Form), refers to Bill Inmon's approach in which it is stated that the data warehouse should be modeled using an E-R model/normalized model.

Dimensional approach
In a dimensional approach, transaction data is partitioned into "facts", which are generally numeric transaction data, and "dimensions", which are the reference information that gives context to the facts. For example, a sales transaction can be broken up into facts such as the number of products ordered and the total price paid for the products, and into dimensions such as order date, customer name, product number, order ship-to and bill-to locations, and salesperson responsible for receiving the order.

A key advantage of a dimensional approach is that the data warehouse is easier for the user to understand and to use. Also, the retrieval of data from the data warehouse tends to operate very quickly. Dimensional structures are easy to understand for business users, because the structure is divided into measurements/facts and context/dimensions. Facts are related to the organization's business processes and operational system whereas the dimensions surrounding them contain context about the measurement (Kimball, Ralph 2008). Another advantage offered by dimensional model is that it does not involve a relational database every time. Thus, this type of modeling technique is very useful for end-user queries in data warehouse.

The model of facts and dimensions can also be understood as a data cube. Where the dimensions are the categorical coordinates in a multi-dimensional cube, the fact is a value corresponding to the coordinates.

The main disadvantages of the dimensional approach are the following:
 * 1) To maintain the integrity of facts and dimensions, loading the data warehouse with data from different operational systems is complicated.
 * 2) It is difficult to modify the data warehouse structure if the organization adopting the dimensional approach changes the way in which it does business.

Normalized approach
In the normalized approach, the data in the data warehouse are stored following, to a degree, database normalization rules. Tables are grouped together by subject areas that reflect general data categories (e.g., data on customers, products, finance, etc.). The normalized structure divides data into entities, which creates several tables in a relational database. When applied in large enterprises the result is dozens of tables that are linked together by a web of joins. Furthermore, each of the created entities is converted into separate physical tables when the database is implemented (Kimball, Ralph 2008). The main advantage of this approach is that it is straightforward to add information into the database. Some disadvantages of this approach are that, because of the number of tables involved, it can be difficult for users to join data from different sources into meaningful information and to access the information without a precise understanding of the sources of data and of the data structure of the data warehouse.

Both normalized and dimensional models can be represented in entity–relationship diagrams as both contain joined relational tables. The difference between the two models is the degree of normalization (also known as Normal Forms). These approaches are not mutually exclusive, and there are other approaches. Dimensional approaches can involve normalizing data to a degree (Kimball, Ralph 2008).

In Information-Driven Business, Robert Hillard proposes an approach to comparing the two approaches based on the information needs of the business problem. The technique shows that normalized models hold far more information than their dimensional equivalents (even when the same fields are used in both models) but this extra information comes at the cost of usability. The technique measures information quantity in terms of information entropy and usability in terms of the Small Worlds data transformation measure.

Bottom-up design
In the bottom-up approach, data marts are first created to provide reporting and analytical capabilities for specific business processes. These data marts can then be integrated to create a comprehensive data warehouse. The data warehouse bus architecture is primarily an implementation of "the bus", a collection of conformed dimensions and conformed facts, which are dimensions that are shared (in a specific way) between facts in two or more data marts.

Top-down design
The top-down approach is designed using a normalized enterprise data model. "Atomic" data, that is, data at the greatest level of detail, are stored in the data warehouse. Dimensional data marts containing data needed for specific business processes or specific departments are created from the data warehouse.

Hybrid design
Data warehouses often resemble the hub and spokes architecture. Legacy systems feeding the warehouse often include customer relationship management and enterprise resource planning, generating large amounts of data. To consolidate these various data models, and facilitate the extract transform load process, data warehouses often make use of an operational data store, the information from which is parsed into the actual data warehouse. To reduce data redundancy, larger systems often store the data in a normalized way. Data marts for specific reports can then be built on top of the data warehouse.

A hybrid (also called ensemble) data warehouse database is kept on third normal form to eliminate data redundancy. A normal relational database, however, is not efficient for business intelligence reports where dimensional modelling is prevalent. Small data marts can shop for data from the consolidated warehouse and use the filtered, specific data for the fact tables and dimensions required. The data warehouse provides a single source of information from which the data marts can read, providing a wide range of business information. The hybrid architecture allows a data warehouse to be replaced with a master data management repository where operational (not static) information could reside.

The data vault modeling components follow hub and spokes architecture. This modeling style is a hybrid design, consisting of the best practices from both third normal form and star schema. The data vault model is not a true third normal form, and breaks some of its rules, but it is a top-down architecture with a bottom up design. The data vault model is geared to be strictly a data warehouse. It is not geared to be end-user accessible, which, when built, still requires the use of a data mart or star schema-based release area for business purposes.

Characteristics
There are basic features that define the data in the data warehouse that include subject orientation, data integration, time-variant, nonvolatile data, and data granularity.

Subject-oriented
Unlike the operational systems, the data in the data warehouse revolves around the subjects of the enterprise. Subject orientation is not database normalization. Subject orientation can be really useful for decision-making. Gathering the required objects is called subject-oriented.

Integrated
The data found within the data warehouse is integrated. Since it comes from several operational systems, all inconsistencies must be removed. Consistencies include naming conventions, measurement of variables, encoding structures, physical attributes of data, and so forth.

Time-variant
While operational systems reflect current values as they support day-to-day operations, data warehouse data represents a long time horizon (up to 10 years) which means it stores mostly historical data. It is mainly meant for data mining and forecasting. (E.g. if a user is searching for a buying pattern of a specific customer, the user needs to look at data on the current and past purchases.)

Nonvolatile
The data in the data warehouse is read-only, which means it cannot be updated, created, or deleted (unless there is a regulatory or statutory obligation to do so).

Aggregation
In the data warehouse process, data can be aggregated in data marts at different levels of abstraction. The user may start looking at the total sale units of a product in an entire region. Then the user looks at the states in that region. Finally, they may examine the individual stores in a certain state. Therefore, typically, the analysis starts at a higher level and drills down to lower levels of details.

Virtualization
With data virtualization, the data used remains in its original locations and real-time access is established to allow analytics across multiple sources creating a virtual data warehouse. This can aid in resolving some technical difficulties such as compatibility problems when combining data from various platforms, lowering the risk of error caused by faulty data, and guaranteeing that the newest data is used. Furthermore, avoiding the creation of a new database containing personal information can make it easier to comply with privacy regulations. However, with data virtualization, the connection to all necessary data sources must be operational as there is no local copy of the data, which is one of the main drawbacks of the approach.

Architecture
The different methods used to construct/organize a data warehouse specified by an organization are numerous. The hardware utilized, software created and data resources specifically required for the correct functionality of a data warehouse are the main components of the data warehouse architecture. All data warehouses have multiple phases in which the requirements of the organization are modified and fine-tuned.

Versus operational system
Operational systems are optimized for the preservation of data integrity and speed of recording of business transactions through use of database normalization and an entity–relationship model. Operational system designers generally follow Codd's 12 rules of database normalization to ensure data integrity. Fully normalized database designs (that is, those satisfying all Codd rules) often result in information from a business transaction being stored in dozens to hundreds of tables. Relational databases are efficient at managing the relationships between these tables. The databases have very fast insert/update performance because only a small amount of data in those tables is affected each time a transaction is processed. To improve performance, older data are usually periodically purged from operational systems.

Data warehouses are optimized for analytic access patterns. Analytic access patterns generally involve selecting specific fields and rarely if ever select *, which selects all fields/columns, as is more common in operational databases. Because of these differences in access patterns, operational databases (loosely, OLTP) benefit from the use of a row-oriented DBMS whereas analytics databases (loosely, OLAP) benefit from the use of a column-oriented DBMS. Unlike operational systems which maintain a snapshot of the business, data warehouses generally maintain an infinite history which is implemented through ETL processes that periodically migrate data from the operational systems over to the data warehouse.

Evolution in organization use
These terms refer to the level of sophistication of a data warehouse:

 * Offline operational data warehouse: Data warehouses in this stage of evolution are updated on a regular time cycle (usually daily, weekly or monthly) from the operational systems and the data is stored in an integrated reporting-oriented database.
 * Offline data warehouse: Data warehouses at this stage are updated from data in the operational systems on a regular basis and the data warehouse data are stored in a data structure designed to facilitate reporting.
 * On-time data warehouse: Online Integrated Data Warehousing represent the real-time Data warehouses stage data in the warehouse is updated for every transaction performed on the source data
 * Integrated data warehouse: These data warehouses assemble data from different areas of business, so users can look up the information they need across other systems.

title: Data warehouse


text:
Disperser

A disperser is a one-sided extractor. Where an extractor requires that every event gets the same probability under the uniform distribution and the extracted distribution, only the latter is required for a disperser. So for a disperser, an event $$A \subseteq \{0,1\}^{m}$$ we have: $$Pr_{U_{m}}[A] > 1 - \epsilon$$

Definition (Disperser): A $$(k, \epsilon)$$-disperser is a function

$$Dis: \{0,1\}^{n}\times \{0,1\}^{d}\rightarrow \{0,1\}^{m}$$

such that for every distribution $$X$$ on $$\{0,1\}^{n}$$ with $$H_{\infty}(X) \geq k$$ the support of the distribution $$Dis(X,U_{d})$$ is of size at least $$(1-\epsilon)2^{m}$$.

Graph theory
An (N, M, D, K, e)-disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than (1 − e)M vertices on the right.

An extractor is a related type of graph that guarantees an even stronger property; every (N, M, D, K, e)-extractor is also an (N, M, D, K, e)-disperser.

Other meanings
A disperser is a high-speed mixing device used to disperse or dissolve pigments and other solids into a liquid.

title: Disperser