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Brandy

Brandy is a liquor produced by distilling wine. Brandy generally contains 35–60% alcohol by volume (70–120 US proof) and is typically consumed as an after-dinner digestif. Some brandies are aged in wooden casks. Others are coloured with caramel colouring to imitate the effect of aging, and some are produced using a combination of aging and colouring. Varieties of wine brandy can be found across the winemaking world. Among the most renowned are Cognac and Armagnac from south-western France.

In a broader sense, the term brandy also denotes liquors obtained from the distillation of pomace (yielding pomace brandy), or mash or wine of any other fruit (fruit brandy). These products are also called eau de vie (literally "water of life" in French).

History
The origins of brandy are tied to the development of distillation. While the process was known in classical times, it was not used for significant beverage production until the 15th century. In the early 16th-century French brandy helped kickstart the cross-Atlantic triangle trade when it took over the central role of the Portuguese fortified wine due to its higher alcohol content and ease of shipping. Canoemen and guards on the African side of the trade were generally paid in brandy. By the late 17th century, rum had replaced brandy as the exchange alcohol of choice in the triangle trade.

Initially, wine was distilled as a preservation method and to make it easier for merchants to transport. It is also thought that wine was originally distilled to lessen the tax, which was assessed by volume. The intent was to add the water removed by distillation back to the brandy shortly before consumption. It was discovered that after being stored in wooden casks, the resulting product improved over the original distilled spirit. In addition to removing water, the distillation process led to the formation and decomposition of numerous aromatic compounds, fundamentally altering the distillate composition from its source. Non-volatile substances such as pigments, sugars, and salts remained behind in the still. As a result, the distillate taste was often quite unlike the sources.

As described in the 1728 edition of Cyclopaedia, the following method was used to distill brandy: "A cucurbit was filled half full of the liquor from which brandy was to be drawn and then raised with a little fire until about one-sixth part was distilled, or until that which falls into the receiver was entirely flammable. This liquor, distilled only once, was called the spirit of wine or brandy. Purified by another distillation (or several more), this was called spirit of wine rectified. The second distillation was made in [a] balneo mariae and in a glass cucurbit, and the liquor was distilled to about one-half the quantity. This was further rectified as long as the operator thought it necessary to produce brandy."

To shorten these several distillations, which were long and troublesome, a chemical instrument was invented that reduced them to a single distillation. A portion was ignited to test the purity of the rectified spirit of wine. The liquor was good if a fire consumed the entire contents without leaving any impurities behind. Another, better test involved putting a little gunpowder in the bottom of the spirit. The liquor was good if the gunpowder could ignite after the spirit was consumed by fire. (Hence the modern "proof" to describe alcohol content.)

As most brandies have been distilled from grapes, the regions of the world producing excellent brandies have roughly paralleled those areas producing grapes for viniculture. At the end of the 19th century, the western European markets, including by extension their overseas empires, were dominated by French and Spanish brandies and eastern Europe was dominated by brandies from the Black Sea region, including Bulgaria, the Crimea, Georgia and Armenia. In 1877, the Ararat brandy brand was established in Yerevan, Armenia. It became one of the top brandy brands over time and during the Yalta Conference, Winston Churchill was so impressed with the Armenian brandy Dvin given to him by Joseph Stalin that he asked for several cases of it to be sent to him each year. Reportedly 400 bottles of Dvin were shipped to Churchill annually. In 1884, David Sarajishvili founded a brandy factory in Tbilisi, Georgia, a crossroads for Turkish, Central Asian, and Persian trade routes and a part of the Russian Empire at the time.

Technology
Except for a few major producers, brandy production and consumption tend to have a regional character, and thus production methods significantly vary. Wine brandy is produced from a variety of grape cultivars. A special selection of cultivars, providing distinct aroma and character, is used for high-quality brandies, while cheaper ones are made from whichever wine is available.

Brandy is made from so-called base wine, which significantly differs from regular table wines. It is made from early grapes to achieve higher acid concentration and lower sugar levels. Base wine generally contains smaller amounts (up to 20 mg/L) of sulphur than regular wines, as it creates undesired copper(II) sulfate in reaction with copper in the pot stills. The yeast sediment produced during the fermentation may or may not be kept in the wine, depending on the brandy style.

Brandy is distilled from the base wine in two phases. First, a large part of water and solids is removed from the base, obtaining so-called "low wine", a concentrated wine with 28–30% ABV. In the second stage, low wine is distilled into brandy. The liquid exits the pot still in three phases, referred to as the "heads", "heart", and "tails", respectively. The first part, the "head", has an alcohol concentration of about 83% (166 US proof) and an unpleasant odour. The weak portion on the end, the "tail", is discarded along with the head, and they are generally mixed with another batch of low wine, thereby entering the distillation cycle again. The middle heart fraction, the richest in aromas and flavours, is preserved for later maturation.

Distillation does not simply enhance the alcohol content of wine. The heat under which the product is distilled and the material of the still (usually copper) cause chemical reactions during distillation. This leads to the formation of numerous new volatile aroma components, changes in relative amounts of aroma components in the wine, and the hydrolysis of components such as esters.

Brandy is usually produced in pot stills (batch distillation), but the column still can also be used for continuous distillation. The distillate obtained in this manner has a higher alcohol concentration (approximately 90% ABV) and is less aromatic. The choice of the apparatus depends on the style of brandy produced. Cognac and South African brandy are examples of brandy produced in batches while many American brandies use fractional distillation in column stills.

Aging
After distillation, the unaged brandy is placed into oak barrels to mature. Usually, brandies with a natural golden or brown colour are aged in oak casks (single-barrel ageing). Some brandies, particularly those from Spain, are aged using the solera system, where the producer changes the barrel each year. After a period of ageing, which depends on the style, class and legal requirements, the mature brandy is mixed with distilled water to reduce alcohol concentration and bottled. Some brandies have caramel colour and sugar added to simulate the appearance of barrel ageing.

Serving
Brandy is traditionally served at room temperature (neat) from a snifter, a wine glass or a tulip glass. When drunk at room temperature, it is often slightly warmed by holding the glass cupped in the palm or by gentle heating. Excessive heating of brandy may cause the alcohol vapour to become too strong, causing its aroma to become overpowering. Brandy-drinkers who like their brandy warmed may ask for the glass to be heated before the brandy is poured.

Brandy may be added to other beverages to make several popular cocktails; these include the Brandy Sour, the Brandy Alexander, the Sidecar, the Brandy Daisy, and the Brandy Old Fashioned.

Anglo-Indian usage has "brandy-pawnee" (brandy with water).

Culinary uses
Brandy is a common deglazing liquid used in making pan sauces for steak and other meat. It creates a more intense flavour in some soups, notably onion soup.

In English Christmas cooking, brandy is a common flavouring in traditional foods such as Christmas cake, brandy butter, and Christmas pudding. It is also commonly used in drinks such as mulled wine and eggnog, drunk during the festive season.

Brandy is used to flambé dishes such as crêpe Suzette and cherries jubilee while serving. Brandy is traditionally poured over a Christmas pudding and set alight before serving. The use of flambé can retain as much as 75% of the alcohol in the brandy.

Historic medical uses
In the 19th century, brandy was often used as medical treatment due to its alleged "stimulating" qualities. It was also used by many European explorers of tropical Africa, who suggested that regular, moderate doses of brandy might help a traveller to cope with fever, depression, and stress. These views fell out of favour in the late nineteenth and early twentieth century, with suggestions that people were using brandy's "medical" qualities as an excuse for social drinking.

Terminology and legal definitions
The term brandy is a shortening of the archaic English brandewine or brandywine, which was derived from the Dutch word brandewijn, itself derived from gebrande wijn, which literally means "burned wine" and whose cognates include brännvin and brennivín. In Germany, the term Branntwein refers to any distilled spirits, while Weinbrand refers specifically to distilled wine from grapes.

In the general colloquial usage of the term, brandy may also be made from pomace and from fermented fruit other than grapes. If a beverage comes from a particular fruit (or multiple fruits) other than exclusively grapes, or from the must of such fruit, it may be referred to as a "fruit brandy" or "fruit spirit" or named using the specific fruit, such as "peach brandy", rather than just generically as "brandy". If pomace is the raw material, the beverage may be called "pomace brandy", "marc brandy", "grape marc", "fruit marc spirit", or "grape marc spirit", "marc" being the pulp residue after the juice has been pressed from the fruit.

Grape pomace brandy may be designated as "grappa" or "grappa brandy". Apple brandy may be referred to as "applejack", although the process of jacking which was originally used in its production involved no distillation. There is also a product called "grain brandy" that is made from grain spirits.

Within particular jurisdictions, specific regulatory requirements regarding the labelling of products identified as brandy exist. For example:

 * In the European Union, there are regulations that require products labelled as brandy, except "grain brandy", to be produced exclusively from the distillation or redistillation of grape-based wine or grape-based "wine fortified for distillation" and aged a minimum of six months in oak.
 * In the US, a brandy that has been produced from other than grape wine must be labelled with a clarifying description of the type of brandy production, such as "peach brandy", "fruit brandy", "dried fruit brandy", or "pomace brandy", and brandy that has not been aged in oak for at least two years must be labelled as "immature".
 * In Canada, the regulations regarding naming conventions for brandy are similar to those of the US (provisions B.02.050–061). According to Canadian food and drug regulations, Brandy shall be a potable alcoholic distillate, or a mixture of potable alcoholic distillates, obtained by the distillation of wine. The minimum specified ageing period is six months in wood, although not necessarily oak (provision B.02.061.2). Caramel, fruit, other botanical substances, flavourings, and flavouring preparations may also be included in a product called brandy (provisions B.02.050–059).

Within the European Union, the German term Weinbrand is legally equivalent to the English term "brandy", but outside the German-speaking countries, it is particularly used to designate brandy from Austria and Germany.

Varieties and brands

 * Most American grape brandy production is situated in California. Popular brands include Christian Brothers, E&J Gallo, Korbel, and Paul Masson.
 * Ararat has been produced since 1877 and comes from the Ararat plain in the southern part of Armenia. Bottles on the market are aged anywhere from 3 to 20 years.
 * In France:
 * Armagnac is made from grapes of the Armagnac region in the southwest of France, Gers, Landes and Lot-et-Garonne. It is single-continuous distilled in a copper still and aged in oak casks from Gascony or Limousin or from the renowned Tronçais Forest in Auvergne. Armagnac was the first distilled spirit in France. Its usage was first mentioned in 1310 by Vital Du Four in a book of medicine recipes. Armagnacs have a specificity: they offer vintage qualities. Popular brands are Darroze, Baron de Sigognac, Larressingle, Delord, Laubade, Gélas and Janneau.
 * Cognac comes from the Cognac region of France, and is double distilled using pot stills. Popular brands include Hine, Martell, Camus, Otard, Rémy Martin, Hennessy, Frapin, Delamain and Courvoisier. The European Union and some other countries legally enforce "Cognac" as the exclusive name for brandy produced and distilled in the Cognac area of France and the name "Armagnac" for brandy from the Gascony area of France. Both must also be made using traditional techniques. Since these are considered "protected designations of origin", a brandy made elsewhere may not be called Cognac in these jurisdictions, even if it was made in an identical manner.
 * Fine is any high-quality brandy, including Cognac and Armagnac but also Fine de Bordeaux, Fine de Bourgogne, and Fine de la Marne.
 * Cyprus brandy differs from other varieties in that its alcohol concentration is only 32% ABV (64 US proof).
 * Greek brandy is distilled from Muscat wine. Mature distillates are made from sun-dried Savatiano, Sultana, and Black Corinth grape varieties blended with an aged Muscat wine.
 * Brandy de Jerez originates from vineyards around Jerez de la Frontera in Andalusia, Spain. It is used in some sherries and is also available as a separate product. It has a protected designation of origin (PDO).
 * Kanyak (or konyak) is a variety from Turkey, whose name is both a variation of "cognac" and means "burn blood" in Turkish, a reference to its use in cold weather.
 * Portuguese Lourinhã region, just north of Lisbon, is one of the few European PDO that produce only brandy (aguardente vínica), together with Cognac, Armagnac and Jerez.
 * In Moldova and Romania, grape brandy is colloquially called coniac, but is officially named Divin in Moldova and Vinars in Romania. After a double distillation, the beverage is usually aged in oak barrels and labelled according to its age (VS is a minimum of 3 years old, VSOP is a minimum of 5 years old, XO is a minimum of 7 years old, and XXO is a minimum of 20 years old).
 * In Russia, brandy was first produced in 1885 at the Kizlyar Brandy Factory according to a recipe brought from France.   Kizlyar brandy is produced according to the classic cognac technology and is one of the most popular beverages in Russia. Also in 2008, the factory restored the status of the Kremlin Suppliers Guild.
 * South African brandies are, by law, made almost exactly as Cognac, using a double distillation process in copper pot stills followed by ageing in oak barrels for a minimum of three years. Because of this, South African brandies are considered very high quality.
 * Italian Stravecchio has been produced since the 1700s in the North of Italy, especially in Emilia-Romagna and Veneto, using grapes that are popular in winemaking such as Sangiovese and Grignolino. Colour, texture and finish resemble those of their French and Spanish counterparts. The most popular brands are Vecchia Romagna, Stravecchio Branca and Stock 84. Northern Italy has also been noted since the Middle Ages for another type of wine spirit, Grappa, which is generally colourless but has some top-shelf varieties called barrique which are aged in oak casks and achieve the same caramel colour as regular brandies. There is a vast production of Stravecchios and Grappas in Italy, with more than 600 large, medium or small distilleries. Ticino, in Italian-speaking Switzerland, is also allowed to produce pomace brandy under the name of Grappa.

Labelling of grades
Brandy has a traditional age grading system, although its use is unregulated outside of Cognac and Armagnac. These indicators can usually be found on the label near the brand name:

 * V.S. ("very special") or ✯✯✯ (three stars) designates a blend in which the youngest brandy has been stored for at least two years in a cask.
 * V.S.O.P. ("very superior old pale"), Reserve or ✯✯✯✯✯ (five stars) designates a blend in which the youngest brandy is stored for at least four years in a cask.
 * XO ("extra old") or Napoléon designates a blend in which the youngest brandy is stored for at least six years.
 * Hors d'âge ('beyond age') is a designation formally equal to XO for Cognac, but for Armagnac it designates brandy that is at least ten years old. In practice, the term is used by producers to market a high-quality product beyond the official age scale.

In the case of Brandy de Jerez, the Consejo Regulador de la Denominacion Brandy de Jerez classifies it according to:
 * Brandy de Jerez Solera: 6 months old.
 * Brandy de Jerez Solera Reserva: one year old.
 * Brandy de Jerez Solera Gran Reserva: three years old.

Russian brandy (traditionally called "Cognac" within the country), as well as brandies from many other post-Soviet states, uses the traditional Russian grading system that is similar to the French one, but extends it significantly:
 * "Three stars" or ✯✯✯ designates the brandy with the youngest component cask-aged for at least two years, analogous to the French V.S.
 * "Four stars" or ✯✯✯✯ is for the blends where the youngest brandy is aged for at least three years.
 * "Five stars" means that the youngest brandy in the blend was aged four years, similar to the French V.S.O.P.
 * КВ/KV ("Aged Cognac") is a designation corresponding to "XO" or "Napoléon", meaning that the youngest spirit in the blend is at least six years old.
 * КВВК/KVVK ("Aged Cognac, Superior Quality") designates the eight-year-old blends and tends to be used only for the highest quality vintages.
 * КС/KS ("Old Cognac"): At least ten years of age for the youngest spirit in the blend (similar to the Armagnac's "Hors d'âge").
 * ОС/OS ("Very Old"): Beyond the French system, designating blends older than 20 years.

title: Brandy


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Carl Maria von Weber

Carl Maria Friedrich Ernst von Weber (c. 18 November 1786 – 5 June 1826) was a German composer, conductor, virtuoso pianist, guitarist, and critic of the early Romantic period. Best known for his operas, he was a crucial figure in the development of German Romantische Oper (German Romantic opera).

Throughout his youth, his father, Franz Anton von Weber, relentlessly moved the family between Hamburg, Salzburg, Freiberg, Augsburg and Vienna. Consequently he studied with many teachers—his father, Johann Peter Heuschkel, Michael Haydn, Giovanni Valesi, Johann Nepomuk Kalcher, and Georg Joseph Vogler—under whose supervision he composed four operas, none of which survive complete. He had a modest output of non-operatic music, which includes two symphonies, two concertos and a concertino for clarinet and orchestra, a bassoon concerto, a horn concertino, two concertos and a Konzertstück for piano and orchestra, piano pieces such as Invitation to the Dance; and many pieces that featured the clarinet, usually written for the virtuoso clarinetist Heinrich Baermann.

His mature operas—Silvana (1810), Abu Hassan (1811), Der Freischütz (1821), Die drei Pintos (comp. 1820–21), Euryanthe (1823), Oberon (1826)—had a major impact on subsequent German composers including Marschner, Meyerbeer, and Wagner; his compositions for piano influenced those of Mendelssohn, Chopin and Liszt. His best known work, Der Freischütz, remains among the most significant German operas.

Childhood
Carl Maria Friedrich Ernst von Weber was born around 18 November 1786 in Eutin, Bishopric of Lübeck. He was the eldest of the three children of Franz Anton von Weber and his second wife, Genovefa Weber, a Viennese singer. He was baptized Catholic on 20 November 1786 with the name Carl Friedrich Ernst; the alternative second name Maria appeared only later. His brother and sister died in infancy. Both parents were Catholic and originally came from the far south of Germany. The "von" was an affectation of his father's, who was not an aristocrat and who claimed descent from a south German noble family which was already extinct at the time.

In April 1779, Franz Anton had been appointed director of the prince-bishopric orchestra, Eutin, which, however, was dissolved in 1781 because of spending cuts. He then took the position of Eutin's municipal music director. Dissatisfied with this position, he resigned in 1787 and founded a theatre company in Hamburg. After a brief stay in Vienna, he joined the theatre company of Johann Friedrich Toscani (husband of Elisabeth Toscani) and Peter Carl Santorini, who performed in Kassel, Marburg, and Hofgeismar. He tried repeatedly to establish a lasting company of his own but had only intermittent success.

Franz Anton's half-brother, Franz Fridolin, married Cäcilia Stamm and had four daughters—Josepha, Aloysia, Constanze, and Sophie—all of whom became notable singers. Wolfgang Amadeus Mozart attempted to woo Aloysia, composing several pieces for her. After she rejected his advances, Mozart went on to marry Constanze; thus Mozart's wife was a cousin of Carl Maria von Weber.

A gifted violinist, Franz Anton had ambitions of turning Weber into a child prodigy like Mozart. Weber was born with a congenital hip disorder and did not begin to walk until he was four. But by then, he was already a capable singer and pianist.

Education
Franz Anton gave Weber a comprehensive education, which was frequently interrupted by the family's moves. In 1796, Weber continued his musical education in Hildburghausen, where he was instructed by the oboist Johann Peter Heuschkel. After moving to Salzburg in autumn 1797, Weber studied from 1798 with Michael Haydn, younger brother of the better-known Joseph Haydn, who agreed to teach Weber free of charge.

His time in Salzburg was overshadowed by the death of his mother Genovefa, who succumbed to tuberculosis on 13 March 1798, and that of his one-year-old sister Antonetta on 29 December 1798 in Munich. After the death of Carl's mother, his paternal aunt Adelaide took over the care of him.

A visit to Joseph Haydn in Vienna, presumably in hope of advanced teaching, was fruitless. In autumn 1798, Weber moved to Munich where he studied singing with Johann Evangelist Wallishauser and composing with Johann Nepomuk Kalcher, who supervised Weber's first opera, Die Macht der Liebe und des Weins (The Power of Love and Wine). Like his other compositions of that period, this opera is lost. Six fughettas for piano of the twelve-year-old Weber were published in Leipzig.

Weber's musical education was extended by a mastering of lithography which he learned in the workshop of Alois Senefelder (the inventor of the process) and Franz Gleißner (autumn 1799). A set of his Variations for the Pianoforte was lithographed by Weber himself.

In 1800, the family moved to Freiberg in Saxony, where Weber, then 14 years old, wrote an opera called Das stumme Waldmädchen (The Silent Forest Maiden). It was produced at the Freiberg and Chemnitz theatres and later in Saint Petersburg (1804), Vienna and Prague (1806). The young Weber also began to publish articles as a music critic, for example in the Leipziger Neue Zeitung in 1801.

In 1801, the family returned to Salzburg, where Weber resumed his studies with Michael Haydn. Weber composed his third opera Peter Schmoll und seine Nachbarn (Peter Schmoll and His Neighbours) of which his teacher approved. After a concert tour in 1802 the Webers returned to Augsburg where it is believed Peter Schmoll premiered.

In mid 1803, Weber continued his studies in Vienna with Abbé Vogler, founder of important music schools in Mannheim, Stockholm, and Darmstadt. Another famous pupil of Vogler in Darmstadt was Jakob Meyer Beer, later known as Giacomo Meyerbeer, who became a close friend of Weber. In letters they addressed each other as "brother".

Early career 1804–1810
Vogler recommended the 17-year-old Weber for the post of Director at the Breslau Opera in 1804; Weber was offered and accepted the post. He sought to reform the Opera by pensioning off older singers, expanding the orchestra, and tackling a more challenging repertoire. His ambitious and dedicated work as director of the orchestra was acknowledged, though his tempi were frequently criticized as too fast. As the daily routine did not leave sufficient time for his own creative work, Weber did not seek to extend his two-year appointment.

After an interlude at the court of Duke Eugen of Württemberg, who resided in Silesia, Weber served from 1807 to 1810 in Stuttgart as private secretary to Duke Ludwig, brother of King Frederick I of Württemberg. Weber's time in Württemberg was plagued with troubles. He fell deeply into debt and became entangled in the financial manipulations of his employer, e.g. the sale of confirmations of ducal service which exempted the purchaser from military service. Weber was arrested and charged with embezzlement and bribery. As he could disprove the allegations, the case was brought under civil law to avoid compromising the de facto manipulator, the brother of the king. Weber agreed to pay the costs (the last payment was made in 1816) and was banished from Württemberg together with his father.

As a sobering side effect, Weber started to keep a diary to list his expenses and correspondence, and make occasional comments on special events.

Weber remained prolific as a composer during this period, writing a quantity of religious music, mainly for the Catholic mass. This, however, earned him the hostility of conservatives working for the re-establishment of traditional chant in liturgy. In his biography of Weber, Warrack notes that Weber was an accomplished guitarist. It was in this year that his first song with guitar accompaniment, "Liebeszauber", was printed. Some of his most original and innovative songs were written during the following years, including "Er an Sie" (1808) and "Was zieht zu deinem Zauberkreise" (1809).

Later career 1810–1826
In 1810, Weber visited several cities throughout Germany; 1811 was a pivotal year in his career when he met and worked with the Munich court clarinetist Heinrich Baermann and composed the Concertino in E♭ Major, Op. 26, J. 109, and the two concerti J. 114 and J. 118 for him; from December 1811 through March 1812, Weber went on tour with Baermann playing the clarinet works, and it was some of the final concerts on this tour that changed public, critical, and royal opinions of Weber's work, and helped him to mount a successful performance of Silvana in Berlin later that year. From 1813 to 1816, he was director of the Opera in Prague; from 1816 to 1817 he worked in Berlin, and from 1817 onwards he was director of the prestigious Opera in Dresden, working hard to establish a German opera, in reaction to the Italian opera which had dominated the European music scene since the 18th century. He was inspired in this endeavour by the ideals of the Sturm und Drang period, and also by the German folk song collection "Des Knaben Wunderhorn" by Arnim and Brentano. It was in 1816 that he wrote the Duet Op. 38 for guitar and piano, possibly inspired by similar works printed in Vienna by such composers as Hummel and Diabelli. On 4 November 1817, he married Caroline Brandt, a singer who created the title role of Silvana. In 1819, he wrote perhaps his most famous piano piece, Invitation to the Dance.

The successful premiere of Der Freischütz on 18 June 1821 in Berlin led to performances all over Europe. On the very morning of the premiere, Weber finished his Konzertstück in F minor for Piano and Orchestra, and he premiered it a week later.

In 1823, Weber composed his first (and only) full-length, through-composed opera Euryanthe to a libretto by Helmina von Chézy, several passages of which (notably the music for the villainous couple Lysiart and Eglantine) anticipate the early, romantic operas of Richard Wagner. In 1824, Weber received an invitation from The Royal Opera, London, to compose and produce Oberon, based on Christoph Martin Wieland's poem of the same name. Weber accepted the invitation, and in 1826 he travelled to England, to finish the work and conduct the premiere on 12 April.

Weber was already suffering from tuberculosis when he visited London. He conducted the premiere and twelve sold-out performances of Oberon in London during April and in May, and despite his rapidly worsening health, he continued to fulfil commitments for private concerts and benefits.

He died in his sleep during the night on 5 June 1826 at the home of his good friend and host Sir George Smart; he was 39 years old. He was buried in London.

Eighteen years later, in December 1844, his remains were transferred to the family burial plot in the Old Catholic Cemetery in Dresden at the side of his youngest son Alexander, who at the age of 19 had died of measles seven weeks before. The simple gravestone, designed by Gottfried Semper, lies against the northern boundary wall. Wagner composed a eulogy "An Weber's Grabe" WWV 72 for the reburial. The piece for male choir a cappella was premiered on 16 December 1844 in Dresden. For this occasion Wagner also composed Funeral Music for Winds after Themes from "Euryanthe" of Weber, WWV 73.

Weber's unfinished comic opera Die drei Pintos (The Three Pintos) was originally given by his widow to Meyerbeer for completion; it was eventually completed by Gustav Mahler, who conducted the first performance in Leipzig on 20 January 1888.

Legacy
Weber's operas Der Freischütz, Euryanthe, and Oberon greatly influenced the development of the Romantische Oper (Romantic opera) in Germany. Der Freischütz came to be regarded as the first German opera, Euryanthe developed the leitmotif technique to an unprecedented degree, while Oberon may have influenced Mendelssohn's music for A Midsummer Night's Dream and, at the same time, revealed Weber's lifelong interest in the music of non-Western cultures. This interest was first manifested in Weber's incidental music for Schiller's translation of Gozzi's Turandot, for which he used a Chinese melody, making him the first Western composer to use an Asian tune that was not of the pseudo-Turkish kind popularized by Mozart and others.

Weber's compositions for clarinet, bassoon, and horn occupy an important place in the musical repertoire. His compositions for the clarinet, which include two concertos, a concertino, a quintet, a duo concertante, and variations on a theme from his opera Silvana, are regularly performed today. His Concertino for Horn and Orchestra requires the performer to simultaneously produce two notes by humming while playing—a technique known as "multiphonics". His bassoon concerto and the Andante e Rondo ungarese (a reworking of a piece originally for viola and orchestra) are also popular with bassoonists.

Weber's contribution to vocal and choral music is also significant. His body of Catholic religious music was highly popular in 19th-century Germany, and he composed one of the earliest song cycles, Die Temperamente beim Verluste der Geliebten ([Four] Temperaments on the Loss of a Lover). Weber was also notable as one of the first conductors to conduct without a piano or violin.

Weber's orchestration has also been highly praised and emulated by later generations of composers—Berlioz referred to him several times in his Treatise on Instrumentation while Debussy remarked that the sound of the Weber orchestra was obtained through the scrutiny of the soul of each instrument.

His operas influenced the work of later opera composers, especially in Germany, such as Marschner, Meyerbeer, and Wagner, as well as several nationalist 19th-century composers such as Glinka. Homage has been paid to Weber by many 20th-century composers, such as Debussy and Stravinsky. Mahler completed Weber's unfinished comic opera Die drei Pintos and made revisions of Euryanthe and Oberon while Hindemith composed the popular Symphonic Metamorphosis of Themes by Carl Maria von Weber, based on Weber's lesser-known keyboard works and the incidental music to Turandot.

Weber also wrote music journalism and was interested in folksong, and learned lithography to engrave his own works.

A virtuoso pianist himself, Weber composed four sonatas, two concertos, and the Konzertstück in F minor (Concert Piece), which inspired composers such as Chopin, Liszt, and Mendelssohn. The Konzertstück provided a new model for the one-movement concerto in several contrasting sections (such as Liszt's, who often played the work), and was acknowledged by Stravinsky as the model for his Capriccio for Piano and Orchestra. Weber's shorter piano pieces, such as the Invitation to the Dance, were later orchestrated by Berlioz, while his Polacca Brillante was later set for piano and orchestra by Liszt. However, Weber's piano music all but disappeared from the repertoire. There are several recordings of the major works for the solo piano, including complete recordings of the piano sonatas and the shorter piano pieces, and there are recordings of the individual sonatas by Claudio Arrau (1st sonata), Alfred Cortot and Emil Gilels (2nd sonata), Sviatoslav Richter (3rd sonata), and Leon Fleisher (4th sonata). The Invitation to the Dance, although better known in Berlioz's orchestration (as part of the ballet music for a Paris production of Der Freischütz), has long been played and recorded by pianists (e.g., by Benno Moiseiwitsch [in Carl Tausig's arrangement]). Invitation to the Dance also served as the thematic basis for Benny Goodman's swing theme song for the radio program Let's Dance.

Books

 * Two volumes.
 * Two volumes.
 * Two volumes.

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

title: Carl Maria von Weber


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Talk:Thematic role

Would a discussion of how ergative and accusative languages emphasize different thematic roles be appropriate in this article? In particular, I'm thinking of the fact that ergative languages tend to emphasize the patient of a verb (by always marking it identically), whereas accusative languages seem to emphasize the agent of a verb. By comparing and contrasting how ergative verbs are handled in these languages, it is possible to illustrate that sometimes a single NP can simultaneously be agent and patient of the verb. For example, using the Basque language ergative marker -ak (or appropriate pronomials), one gets:

This illustration begs the question of whether, Subject/Object/Verb typology is appropriate for ergative languages, or whether both types of languages might be better categorized by the sequences of the Agent and Patient relative to the verb. Thus, English would be a Agent-Verb-Patient language, with Agent prominence; Basque would be a Patient-Agent-Verb language, with Patient prominence.
 * The Subject/Object/Verb typology is perfectly apropriate to describe syntax. The theta roles - or thematic roles - are appropriate for describing semantics or meaning.  Which categorisation is better depends entirely on one's purpose.  A joint description of both the semantics and the syntax of an utterance would use both typologies.yoyo 12:34, 25 February 2006 (UTC)

Thematic role and theta role
Do we really need two separate articles - thematic role and theta role? Boraczek 21:42, 5 Dec 2004 (UTC)
 * No, they are about the same topic. Hirzel 10:18, 25 Jun 2005 (UTC)
 * Thematic role is a semantic idea, Theta role is syntactic. --April Arcus 22:18, 10 November 2005 (UTC)
 * Theta role is explicitly described in its article Theta role as a semantic role! Agreed that both articles have similar referents.  The thematic role article includes several roles that are glossed over in the theta role article.  Apparently theta role is the commoner term in linguistics circles, though I feel that thematic role suggests its meaning better. yoyo 12:34, 25 February 2006 (UTC)
 * Theta roles or equivalently θ-roles are the terms used for thematic roles in the context of Government and Binding Theory, but essentially they are the same thing. I say the articles should be merged, but perhaps some point should be made about how GB theory emphasizes the idea of biuniqueness in thematic roles, which has applications for syntax.  Thematic roles or theta roles have to do with both syntax and semantics, since they help to correlate syntactic behavior (like the agent tending to the the subject) with semantic properties of noun phrases (like agents tending to have volition and perception).  According to Dowty (1991) thematic roles are part of the syntax-semantics interface. haplo 21:06, 8 May 2006 (UTC)

 * Theta role is the syntactic reflection of the semantic concept of thematic role and as such these articles definitely should be merged. Theta role is the more common term, so that's the place to merge with. — mark ✎ 08:24, 22 April 2006 (UTC)

In case of a merge, I really would suggest to keep things as clear as possible: the theta role undoubtly is a syntactic concept. The theme, however, is quite different from that if you take the next step within linguistic hierarchy, which is semantics. And if one goes even one step further into Functional Grammar (which is mostly Pragmatics) and its notion of the stratum of lexicogrammar, the common aspects of "theta role" (which is unknown to Functional Grammar) and "theme" (which is a topic of utmost importance for FG) vanish almost into nonexistence. Although I can understand the desire to merge both articles, I do not think it would be scientifically appropriate. Limbonic 20:59, 27 May 2006 (UTC)

 * I think everyone here is not strongly opposed to merging, and at most have reservations. My opinion is that the Thematic role page should be merged into the Theta role page. These two pages currently contain strikingly redundant information, with no explaination of the difference--having two pages is confusing! This page mostly seems to be an extention of the list of roles in Theta role, which could easily be merged into that section. If there is a subtle difference between theta roles and thematic roles, it needs to be explained in terms that can be understood by non-experts, perhaps in a short "Theta roles vs. Thematic roles" section. I don't think that the difference between thematic role and theta role is enough to ever generate a unique Thematic role page larger than a stub (after redundant information is removed of course). I would love to help merge these pages, but I'm afraid to do so as I am not an expert here!--gwc 22:06, 15 January 2007 (UTC)

 * I agree that the somewhat confusing distinction is not made clear enough in either article, but here is an explanation from Andrew Carnie that might help somebody out who is brave enough to give it a try: "There is not a one-to-one relationship between thematic relations and arguments.  However, linguists have a special construct called a 'theta role,' that does map one-to-one with arguments.  Theta roles are bundles of thematic relations that cluster on one argument.  In [an example he posited earlier], 'Jason' gets two thematic relations (agent and source), but only one theta role (the one that contains the agent and source thematic relations).  Somewhat confusingly, syntacticians often refer to particular theta roles by the most prominent thematic relation that they contain.  So you might hear a syntactician refer to the 'agent theta role' of 'Jason.'  Strictly speaking, this is incorrect:  Agent refers to a thematic relation, whereas the theta role is a bundle of thematic relations."  68.100.79.249 11:31, 19 January 2007 (UTC) Edris Qarghah

Experiencer
According to the article on Theta role "the experiencer (S) is someone or a thing who experiences some state. Thus, in Jack fell asleep, Jack is the experiencer. This is because Jack is not an agent, in that he did not 'fall himself asleep'". This suggests that the experiencer works as a subject (at least in English). According to this article however the experiencer is someone who "receives sensory or emotional input (eg. The smell of lillies filled Jennifer's nostrils)". This definition does not suggest that the experiencer is a subject. Is it possible that the writer has over-interpritted the word experiencer? The definition in the Theta role article makes more sense. I'm new to this so I don't want to change anything to hastily. Starylon 18:10, 6 April 2006 (UTC)

 * Experiencer as a thematic relation does not correlate directly to a subject or object position. It is, rather, a semantic question of the association of an argument with a predicate.  In the case of experiencer, the argument has to feel or perceive what is predicated, it's position in the sentence is irrelevant.  Here are a few examples of this from [Andrew Carnie]:  "Keziah likes cookies."  "Becki saw the eclipse."  "Syntax frightens Jim."  Note that in the last example Syntax, as a non-volitional agent, is the active inducer of the experience of fright on Jim.  The verb "frighten" takes two arguments, one, in an object position, has the dominant thematic relation of experiencer.  68.100.79.249 11:50, 19 January 2007 (UTC) Edris Qarghah

Incroguent definition/usage.
i do not think i would say that a role IS a relationship. a role IS PART OF a relationship, but it is not itself the relationship. for example, "Experiencer" is a role, but "Experienced/Experiences/Will Experience" is a relationship.

if you agree, then you should also agree that the Thematic Role shouldn't be defined as "a relationship", as it is in the first sentence, since Experiencer is an example of a Thematic Role, & Experiencer isn't a relationship.

if you disagree, then you should be willing to begin a campaign to rename "Thematic Roles" to "Thematic Relationships".

the lesson is, while roles are related to relationships, they are different.

Factotum 09:54, 8 June 2006 (UTC)

 * I agree that there is a distinction between role and relation, though I fear that your evaluation of it misses the mark a bit. "Thematic Roles" should be renamed "Thematic Relations."  Experiencer is a descriptor of the semantic relationship between the argument and the predicate, namely, an experiencer is an argument that experiences the action depicted in the predicate.  Role implies a position with respect to a specific verb and subsequently a particular type of syntactic structure.  This is part of the distinction between thematic relations and theta roles that really needs to be elucidated.  68.100.79.249 12:02, 19 January 2007 (UTC) Edris Qarghah

title: Talk:Thematic role


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Schumann (disambiguation)

Robert Schumann (1810–1856) was a German composer, music director and publisher of a trade paper.

Schumann or Schuman may also refer to:
 * Schumann (surname), a surname and list of people with names Schumann or Schuman
 * Shuman (surname), list of people with the name
 * Schuman roundabout, an area of Brussels
 * Schuman railway station
 * 4003 Schumann, an asteroid
 * Robert Schuman University, named for French politician Robert Schuman
 * Robert Schumann Hochschule, a university for music and media in Düsseldorf

title: Schumann (disambiguation)


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Lp space

In mathematics, the $L^{p}$ spaces are function spaces defined using a natural generalization of the $p$-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.

$L^{p}$ spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.

Statistics
In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of $$L^p$$ metrics, and measures of central tendency can be characterized as solutions to variational problems.

In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the $L^1$ norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared $$L^2$$ norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero). Elastic net regularization uses a penalty term that is a combination of the $$L^1$$ norm and the squared $$L^2$$ norm of the parameter vector.

Hausdorff–Young inequality
The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps $$L^p(\Reals)$$ to $$L^q(\Reals)$$ (or $$L^p(\mathbf{T})$$ to $$\ell^q$$) respectively, where $$1 \leq p \leq 2$$ and $$\tfrac{1}{p} + \tfrac{1}{q} = 1.$$ This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.

By contrast, if $$p > 2,$$ the Fourier transform does not map into $$L^q.$$

Hilbert spaces
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces $$L^2$$ and $$\ell^2$$ are both Hilbert spaces. In fact, by choosing a Hilbert basis $$E,$$ i.e., a maximal orthonormal subset of $$L^2$$ or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to $$\ell^2(E)$$ (same $$E$$ as above), i.e., a Hilbert space of type $$\ell^2.$$

The $p$-norm in finite dimensions

The Euclidean length of a vector $$x = (x_1, x_2, \dots, x_n)$$ in the $$n$$-dimensional real vector space $$\Reals^n$$ is given by the Euclidean norm: $$\|x\|_2 = \left({x_1}^2 + {x_2}^2 + \dotsb + {x_n}^2\right)^{1/2}.$$

The Euclidean distance between two points $$x$$ and $$y$$ is the length $$\|x - y\|_2$$ of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of $$p$$-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

Definition
For a real number $$p \geq 1,$$ the $$p$$-norm or $$L^p$$-norm of $$x$$ is defined by $$\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb + |x_n|^p\right)^{1/p}.$$ The absolute value bars can be dropped when $$p$$ is a rational number with an even numerator in its reduced form, and $$x$$ is drawn from the set of real numbers, or one of its subsets.

The Euclidean norm from above falls into this class and is the $$2$$-norm, and the $$1$$-norm is the norm that corresponds to the rectilinear distance.

The $$L^\infty$$-norm or maximum norm (or uniform norm) is the limit of the $$L^p$$-norms for $$p \to \infty.$$ It turns out that this limit is equivalent to the following definition: $$\|x\|_\infty = \max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}$$

See $L$-infinity.

For all $$p \geq 1,$$ the $$p$$-norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that: Abstractly speaking, this means that $$\Reals^n$$ together with the $$p$$-norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the $$L^p$$-space over $$\{1, 2, \ldots, n\}.$$
 * only the zero vector has zero length,
 * the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
 * the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).

Relations between $p$-norms
The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm: $$\|x\|_2 \leq \|x\|_1 .$$

This fact generalizes to $$p$$-norms in that the $$p$$-norm $$\|x\|_p$$ of any given vector $$x$$ does not grow with $$p$$:

For the opposite direction, the following relation between the $$1$$-norm and the $$2$$-norm is known: $$\|x\|_1 \leq \sqrt{n} \|x\|_2 ~.$$

This inequality depends on the dimension $$n$$ of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

In general, for vectors in $$\Complex^n$$ where $$0 < r < p:$$ $$\|x\|_p \leq \|x\|_r \leq n^{\frac{1}{r} - \frac{1}{p}} \|x\|_p ~.$$

This is a consequence of Hölder's inequality.

When $0 < p < 1$
In $$\Reals^n$$ for $$n > 1,$$ the formula $$\|x\|_p = \left(|x_1|^p + |x_2| ^p + \cdots + |x_n|^p\right)^{1/p}$$ defines an absolutely homogeneous function for $$0 < p < 1;$$ however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the formula $$|x_1|^p + |x_2|^p + \dotsb + |x_n|^p$$ defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm, though, which is homogeneous of degree $$p.$$

Hence, the function $$d_p(x, y) = \sum_{i=1}^n |x_i - y_i|^p$$ defines a metric. The metric space $$(\Reals^n, d_p)$$ is denoted by $$\ell_n^p.$$

Although the $$p$$-unit ball $$B_n^p$$ around the origin in this metric is "concave", the topology defined on $$\Reals^n$$ by the metric $$B_p$$ is the usual vector space topology of $$\Reals^n,$$ hence $$\ell_n^p$$ is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of $$\ell_n^p$$ is to denote by $$C_p(n)$$ the smallest constant $$C$$ such that the scalar multiple $$C \, B_n^p$$ of the $$p$$-unit ball contains the convex hull of $$B_n^p,$$ which is equal to $$B_n^1.$$ The fact that for fixed $$p < 1$$ we have $$C_p(n) = n^{\tfrac{1}{p} - 1} \to \infty, \quad \text{as } n \to \infty$$ shows that the infinite-dimensional sequence space $$\ell^p$$ defined below, is no longer locally convex.

When $p = 0$
There is one $$\ell_0$$ norm and another function called the $$\ell_0$$ "norm" (with quotation marks).

The mathematical definition of the $$\ell_0$$ norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm $$(x_n) \mapsto \sum_n 2^{-n} \frac{|x_n|}{1 +|x_n|},$$ which is discussed by Stefan Rolewicz in Metric Linear Spaces. The $$\ell_0$$-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the $$\ell_0$$ "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector $$x.$$ Many authors abuse terminology by omitting the quotation marks. Defining $0^0 = 0,$ the zero "norm" of $$x$$ is equal to $$|x_1|^0 + |x_2|^0 + \cdots + |x_n|^0 .$$

This is not a norm because it is not homogeneous. For example, scaling the vector $$x$$ by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.

The sequence space $p$
The $$p$$-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space $$\ell^p.$$This contains as special cases:
 * $$\ell^1,$$ the space of sequences whose series are absolutely convergent,
 * $$\ell^2,$$ the space of square-summable sequences, which is a Hilbert space, and
 * $$\ell^\infty,$$ the space of bounded sequences.

The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: $$\begin{align} & (x_1, x_2, \ldots, x_n, x_{n+1},\ldots)+(y_1, y_2, \ldots, y_n, y_{n+1},\ldots) \\ = {} & (x_1+y_1, x_2+y_2, \ldots, x_n+y_n, x_{n+1}+y_{n+1},\ldots), \\[6pt] & \lambda \cdot \left (x_1, x_2, \ldots, x_n, x_{n+1},\ldots \right) \\ = {} & (\lambda x_1, \lambda x_2, \ldots, \lambda x_n, \lambda x_{n+1},\ldots). \end{align}$$

Define the $$p$$-norm: $$\|x\|_p = \left(|x_1|^p + |x_2|^p + \cdots +|x_n|^p + |x_{n+1}|^p + \cdots\right)^{1/p}$$

Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, $$(1, 1, 1, \ldots),$$ will have an infinite $$p$$-norm for $$1 \leq p < \infty.$$ The space $$\ell^p$$ is then defined as the set of all infinite sequences of real (or complex) numbers such that the $$p$$-norm is finite.

One can check that as $$p$$ increases, the set $$\ell^p$$ grows larger. For example, the sequence $$\left(1, \frac{1}{2}, \ldots, \frac{1}{n}, \frac{1}{n+1}, \ldots\right)$$ is not in $$\ell^1,$$ but it is in $$\ell^p$$ for $$p > 1,$$ as the series $$1^p + \frac{1}{2^p} + \cdots + \frac{1}{n^p} + \frac{1}{(n+1)^p} + \cdots,$$ diverges for $$p = 1$$ (the harmonic series), but is convergent for $$p > 1.$$

One also defines the $$\infty$$-norm using the supremum: $$\|x\|_\infty = \sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \ldots)$$ and the corresponding space $$\ell^\infty$$ of all bounded sequences. It turns out that $$\|x\|_\infty = \lim_{p \to \infty} \|x\|_p$$ if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider $$\ell^p$$ spaces for $$1 \leq p \leq \infty.$$

The $$p$$-norm thus defined on $$\ell^p$$ is indeed a norm, and $$\ell^p$$ together with this norm is a Banach space. The fully general $$L^p$$ space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions. An integral instead of a sum is used to define the $$p$$-norm.

General ℓp-space
In complete analogy to the preceding definition one can define the space $$\ell^p(I)$$ over a general index set $$I$$ (and $$1 \leq p < \infty$$) as $$\ell^p(I) = \left\{(x_i)_{i\in I} \in \mathbb{K}^I : \sum_{i \in I} |x_i|^p < +\infty\right\},$$ where convergence on the right means that only countably many summands are nonzero (see also Unconditional convergence). With the norm $$\|x\|_p = \left(\sum_{i\in I} |x_i|^p\right)^{1/p}$$ the space $$\ell^p(I)$$ becomes a Banach space. In the case where $$I$$ is finite with $$ n$$ elements, this construction yields $$\Reals^n$$ with the $$p$$-norm defined above. If $$I$$ is countably infinite, this is exactly the sequence space $$\ell^p$$ defined above. For uncountable sets $$I$$ this is a non-separable Banach space which can be seen as the locally convex direct limit of $$\ell^p$$-sequence spaces.

For $$p = 2,$$ the $$\|\,\cdot\,\|_2$$-norm is even induced by a canonical inner product $$\langle \,\cdot,\,\cdot\rangle,$$ called the , which means that $$\|\mathbf{x}\|_2 = \sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}$$ holds for all vectors $$\mathbf{x}.$$ This inner product can expressed in terms of the norm by using the polarization identity. On $$\ell^2,$$ it can be defined by $$\langle \left(x_i\right)_{i}, \left(y_n\right)_{i} \rangle_{\ell^2} ~=~ \sum_i x_i \overline{y_i}$$ while for the space $$L^2(X, \mu)$$ associated with a measure space $$(X, \Sigma, \mu),$$ which consists of all square-integrable functions, it is $$\langle f, g \rangle_{L^2} = \int_X f(x) \overline{g(x)}\, \mathrm dx.$$

Now consider the case $$p = \infty.$$ Define $$\ell^\infty(I)=\{x\in \mathbb K^I : \sup\operatorname{range}|x|<+\infty\},$$ where for all $$x$$ $$\|x\|_\infty\equiv\inf\{C \in \Reals_{\geq 0}:|x_i| \leq C\text{ for all } i \in I\} = \begin{cases}\sup\operatorname{range}|x|&\text{if } X\neq\varnothing,\\0&\text{if } X=\varnothing.\end{cases}$$

The index set $$I$$ can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space $$\ell^p(I)$$ is just a special case of the more general $$L^p$$-space (defined below).

Lp spaces and Lebesgue integrals
An $$L^p$$ space may be defined as a space of measurable functions for which the $$p$$-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let $$(S, \Sigma, \mu)$$ be a measure space and $$1 \leq p \leq \infty.$$ When $$p \neq \infty$$, consider the set $$\mathcal{L}^p(S,\, \mu)$$ of all measurable functions $$f$$ from $$S$$ to $$\Complex$$ or $$\Reals$$ whose absolute value raised to the $$p$$-th power has a finite integral, or in symbols: $$\|f\|_p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\int_S |f|^p\;\mathrm{d}\mu\right)^{1/p} < \infty.$$

To define the set for $$p = \infty,$$ recall that two functions $$f$$ and $$g$$ defined on $$S$$ are said to be, written , if the set $$\{s \in S : f(s) \neq g(s)\}$$ is measurable and has measure zero. Similarly, a measurable function $$f$$ (and its absolute value) is (or )  by a real number $$C,$$ written, if the (necessarily) measurable set $$\{s \in S : |f(s)| > C\}$$ has measure zero. The space $$\mathcal{L}^\infty(S,\mu)$$ is the set of all measurable functions $$f$$ that are bounded almost everywhere (by some real $$C$$) and $$\|f\|_\infty$$ is defined as the infimum of these bounds: $$\|f\|_\infty ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \inf \{C \in \Reals_{\geq 0} : |f(s)| \leq C \text{ for almost every } s\}.$$ When $$\mu(S) \neq 0$$ then this is the same as the essential supremum of the absolute value of $$f$$: $$\|f\|_\infty ~=~ \begin{cases}\operatorname{esssup}|f| & \text{if } \mu(S) > 0,\\ 0 & \text{if } \mu(S) = 0.\end{cases}$$

For example, if $$f$$ is a measurable function that is equal to $$0$$ almost everywhere then $$\|f\|_p = 0$$ for every $$p$$ and thus $$f \in \mathcal{L}^p(S,\, \mu)$$ for all $$p.$$

For every positive $$p,$$ the value under $$\|\,\cdot\,\|_p$$ of a measurable function $$f$$ and its absolute value $$|f| : S \to [0, \infty]$$ are always the same (that is, $$\|f\|_p = \||f|\|_p$$ for all $$p$$) and so a measurable function belongs to $$\mathcal{L}^p(S,\, \mu)$$ if and only if its absolute value does. Because of this, many formulas involving $$p$$-norms are stated only for non-negative real-valued functions. Consider for example the identity $$\|f\|_p^r = \|f^r\|_{p/r},$$ which holds whenever $$f \geq 0$$ is measurable, $$r > 0$$ is real, and $$0 < p \leq \infty$$ (here $$\infty / r \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \infty$$ when $$p = \infty$$). The non-negativity requirement $$f \geq 0$$ can be removed by substituting $$|f|$$ in for $$f,$$ which gives $$\|\,|f|\,\|_p^r = \|\,|f|^r\,\|_{p/r}.$$ Note in particular that when $$p = r$$ is finite then the formula $$\|f\|_p^p = \||f|^p\|_1$$ relates the $$p$$-norm to the $$1$$-norm.

Seminormed space of $$p$$-th power integrable functions

Each set of functions $$\mathcal{L}^p(S,\, \mu)$$ forms a vector space when addition and scalar multiplication are defined pointwise. That the sum of two $$p$$-th power integrable functions $$f$$ and $$g$$ is again $$p$$-th power integrable follows from $\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right),$ although it is also a consequence of Minkowski's inequality $$\|f + g\|_p \leq \|f\|_p + \|g\|_p$$ which establishes that $$\|\cdot\|_p$$ satisfies the triangle inequality for $$1 \leq p \leq \infty$$ (the triangle inequality does not hold for $$0 < p < 1$$). That $$\mathcal{L}^p(S,\, \mu)$$ is closed under scalar multiplication is due to $$\|\cdot\|_p$$ being absolutely homogeneous, which means that $$\|s f\|_p = |s| \|f\|_p$$ for every scalar $$s$$ and every function $$f.$$

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus $$\|\cdot\|_p$$ is a seminorm and the set $$\mathcal{L}^p(S,\, \mu)$$ of $$p$$-th power integrable functions together with the function $$\|\cdot\|_p$$ defines a seminormed vector space. In general, the seminorm $$\|\cdot\|_p$$ is not a norm because there might exist measurable functions $$f$$ that satisfy $$\|f\|_p = 0$$ but are not equal to $$0$$ ($$\|\cdot\|_p$$ is a norm if and only if no such $$f$$ exists).

Zero sets of $$p$$-seminorms

If $$f$$ is measurable and equals $$0$$ a.e. then $$\|f\|_p = 0$$ for all positive $$p \leq \infty.$$ On the other hand, if $$f$$ is a measurable function for which there exists some $$0 < p \leq \infty$$ such that $$\|f\|_p = 0$$ then $$f = 0$$ almost everywhere. When $$p$$ is finite then this follows from the $$p = 1$$ case and the formula $$\|f\|_p^p = \||f|^p\|_1$$ mentioned above.

Thus if $$p \leq \infty$$ is positive and $$f$$ is any measurable function, then $$\|f\|_p = 0$$ if and only if $$f = 0$$ almost everywhere. Since the right hand side ($$f = 0$$ a.e.) does not mention $$p,$$ it follows that all $$\|\cdot\|_p$$ have the same zero set (it does not depend on $$p$$). So denote this common set by $$\mathcal{N} \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{f : f = 0 \ \mu\text{-almost everywhere} \} = \{f \in \mathcal{L}^p(S,\, \mu) : \|f\|_p = 0\} \qquad \forall \ p.$$ This set is a vector subspace of $$\mathcal{L}^p(S,\, \mu)$$ for every positive $$p \leq \infty.$$

Quotient vector space

Like every seminorm, the seminorm $$\|\cdot\|_p$$ induces a norm (defined shortly) on the canonical quotient vector space of $$\mathcal{L}^p(S,\, \mu)$$ by its vector subspace $\mathcal{N} = \{f \in \mathcal{L}^p(S,\, \mu) : \|f\|_p = 0\}.$ This normed quotient space is called and it is the subject of this article. We begin by defining the quotient vector space.

Given any $$f \in \mathcal{L}^p(S,\, \mu),$$ the coset $$f + \mathcal{N} \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{f + h : h \in \mathcal{N}\}$$ consists of all measurable functions $$g$$ that are equal to $$f$$ almost everywhere. The set of all cosets, typically denoted by $$\mathcal{L}^p(S, \mu) / \mathcal{N} \stackrel{\scriptscriptstyle\text{def}}{=} \{f + \mathcal{N} : f \in \mathcal{L}^p(S, \mu)\},$$ forms a vector space with origin $$0 + \mathcal{N} = \mathcal{N}$$ when vector addition and scalar multiplication are defined by $$(f + \mathcal{N}) + (g + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (f + g) + \mathcal{N}$$ and $$s (f + \mathcal{N}) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; (s f) + \mathcal{N}.$$ This particular quotient vector space will be denoted by $$L^p(S,\, \mu) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \mathcal{L}^p(S, \mu) / \mathcal{N}.$$

Two cosets are equal $$f + \mathcal{N} = g + \mathcal{N}$$ if and only if $$g \in f + \mathcal{N}$$ (or equivalently, $$f - g \in \mathcal{N}$$), which happens if and only if $$f = g$$ almost everywhere; if this is the case then $$f$$ and $$g$$ are identified in the quotient space.

The $$p$$-norm on the quotient vector space

Given any $$f \in \mathcal{L}^p(S,\, \mu),$$ the value of the seminorm $$\|\cdot\|_p$$ on the coset $$f + \mathcal{N} = \{f + h : h \in \mathcal{N}\}$$ is constant and equal to $$\|f\|_p;$$ denote this unique value by $$\|f + \mathcal{N}\|_p,$$ so that: $$\|f + \mathcal{N}\|_p \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \|f\|_p.$$ This assignment $$f + \mathcal{N} \mapsto \|f + \mathcal{N}\|_p$$ defines a map, which will also be denoted by $$\|\cdot\|_p,$$ on the quotient vector space $$L^p(S, \mu) \stackrel{\scriptscriptstyle\text{def}}{=} \mathcal{L}^p(S, \mu) / \mathcal{N} ~=~ \{f + \mathcal{N} : f \in \mathcal{L}^p(S, \mu)\}.$$ This map is a norm on $$L^p(S, \mu)$$ called the. The value $$\|f + \mathcal{N}\|_p$$ of a coset $$f + \mathcal{N}$$ is independent of the particular function $$f$$ that was chosen to represent the coset, meaning that if $$\mathcal{C} \in L^p(S, \mu)$$ is any coset then $$\|\mathcal{C}\|_p = \|f\|_p$$ for every $$f \in \mathcal{C}$$ (since $$\mathcal{C} = f + \mathcal{N}$$ for every $$f \in \mathcal{C}$$).

The Lebesgue $$L^p$$ space

The normed vector space $$\left(L^p(S, \mu), \|\cdot\|_p\right)$$ is called or the  of $$p$$-th power integrable functions and it is a Banach space for every $$1 \leq p \leq \infty$$ (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem). When the underlying measure space $$S$$ is understood then $$L^p(S, \mu)$$ is often abbreviated $$L^p(\mu),$$ or even just $$L^p.$$ Depending on the author, the subscript notation $$L_p$$ might denote either $$L^p(S, \mu)$$ or $$L^{1/p}(S, \mu).$$

If the seminorm $$\|\cdot\|_p$$ on $$\mathcal{L}^p(S,\, \mu)$$ happens to be a norm (which happens if and only if $$\mathcal{N} = \{0\}$$) then the normed space $$\left(\mathcal{L}^p(S,\, \mu), \|\cdot\|_p\right)$$ will be linearly isometrically isomorphic to the normed quotient space $$\left(L^p(S, \mu), \|\cdot\|_p\right)$$ via the canonical map $$g \in \mathcal{L}^p(S,\, \mu) \mapsto \{g\}$$ (since $$g + \mathcal{N} = \{g\}$$); in other words, they will be, up to a linear isometry, the same normed space and so they may both be called "$$L^p$$ space".

The above definitions generalize to Bochner spaces.

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of $$\mathcal{N}$$ in $$L^p.$$ For $$L^\infty,$$ however, there is a theory of lifts enabling such recovery.

Special cases
Similar to the $$\ell^p$$ spaces, $$L^2$$ is the only Hilbert space among $$L^p$$ spaces. In the complex case, the inner product on $$L^2$$ is defined by $$\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x)$$

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in $$L^2$$ are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral.

If we use complex-valued functions, the space $$L^\infty$$ is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of $$L^\infty$$ defines a bounded operator on any $$L^p$$ space by multiplication.

For $$1 \leq p \leq \infty$$ the $$\ell^p$$ spaces are a special case of $$L^p$$ spaces, when $$S = \mathbb{N})$$ consists of the natural numbers and $$\mu$$ is the counting measure on $$\mathbb{N}.$$More generally, if one considers any set $$S$$ with the counting measure, the resulting $$L^p$$ space is denoted $$\ell^p(S).$$ For example, the space $$\ell^p(\mathbb{Z})$$is the space of all sequences indexed by the integers, and when defining the $$p$$-norm on such a space, one sums over all the integers. The space $$\ell^p(n),$$ where $$n$$ is the set with $$n$$ elements, is $$\Reals^n$$ with its $$p$$-norm as defined above. As any Hilbert space, every space $$L^2$$ is linearly isometric to a suitable $$\ell^2(I),$$ where the cardinality of the set $$I$$ is the cardinality of an arbitrary Hilbertian basis for this particular $$L^2.$$

Properties of Lp spaces
As in the discrete case, if there exists $$q < \infty$$ such that $$f \in L^\infty(S, \mu) \cap L^q(S, \mu),$$ then $$\|f\|_\infty = \lim_{p \to \infty}\|f\|_p.$$

Hölder's inequality

Suppose $$p, q, r \in [1, \infty]$$ satisfy $$\tfrac{1}{p} + \tfrac{1}{q} = \tfrac{1}{r}$$ (where $$\tfrac{1}{\infty} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ 0$$). If $$f \in L^p(S, \mu)$$ and $$g \in L^q(S, \mu)$$ then $$f g \in L^r(S, \mu)$$ and $$\|f g\|_r ~\leq~ \|f\|_p \, \|g\|_q.$$

This inequality, called Hölder's inequality, is in some sense optimal since if $$r = 1$$ (so $$\tfrac{1}{p} + \tfrac{1}{q} = 1$$) and $$f$$ is a measurable function such that $$\sup_{\|g\|_q \leq 1} \, \int_S |f g| \, \mathrm{d} \mu ~<~ \infty$$ where the supremum is taken over the closed unit ball of $$L^q(S, \mu),$$ then $$f \in L^p(S, \mu)$$ and $$\|f\|_p ~=~ \sup_{\|g\|_q \leq 1} \, \int_S f g \, \mathrm{d} \mu.$$

Minkowski inequality

Minkowski inequality, which states that $$\|\cdot\|_p$$ satisfies the triangle inequality, can be generalized: If the measurable function $$F : M \times N \to \Reals$$ is non-negative (where $$(M, \mu)$$ and $$(N, \nu)$$ are measure spaces) then for all $$1 \leq p \leq q \leq \infty,$$ $$\left\|\left\|F(\,\cdot, n)\right\|_{L^p(M, \mu)}\right\|_{L^q(N, \nu)} ~\leq~ \left\|\left\|F(m, \cdot)\right\|_{L^q(N, \nu)}\right\|_{L^p(M, \mu)} \ .$$

Atomic decomposition
If $$1 \leq p < \infty$$ then every non-negative $$f \in L^p(\mu)$$ has an, meaning that there exist a sequence $$(r_n)_{n \in \Z}$$ of non-negative real numbers and a sequence of non-negative functions $$(f_n)_{n \in \Z},$$ called , whose supports $$\left(\operatorname{supp} f_n\right)_{n \in \Z}$$ are pairwise disjoint sets of measure $$\mu\left(\operatorname{supp} f_n\right) \leq 2^{n+1},$$ such that $$f ~=~ \sum_{n \in \Z} r_n \, f_n \, ,$$ and for every integer $$n \in \Z,$$ $$\|f_n\|_\infty ~\leq~ 2^{-\tfrac{n}{p}} \, ,$$ and $$\tfrac{1}{2} \|f\|_p^p ~\leq~ \sum_{n \in \Z} r_n^p ~\leq~ 2 \|f\|^p_p \, ,$$ and where moreover, the sequence of functions $$(r_n f_n)_{n \in \Z}$$ depends only on $$f$$ (it is independent of $$p$$). These inequalities guarantee that $$\|f_n\|_p^p \leq 2$$ for all integers $$n$$ while the supports of $$(f_n)_{n \in \Z}$$ being pairwise disjoint implies $$\|f\|_p^p ~=~ \sum_{n \in \Z} r_n^p \, \|f_n\|^p_p \, .$$

An atomic decomposition can be explicitly given by first defining for every integer $$n \in \Z,$$ $$t_n = \inf \{t \in \Reals : \mu(f > t) < 2^n\}$$ (this infimum is attained by $$t_n;$$ that is, $$\mu(f > t_n) < 2^n$$ holds) and then letting $$r_n ~=~ 2^{n/p} \, t_n ~ \text{ and } \quad f_n ~=~ \frac{f}{r_n} \, \mathbf{1}_{(t_{n+1} < f \leq t_n)}$$ where $$\mu(f > t) = \mu(\{s : f(s) > t\})$$ denotes the measure of the set $$(f > t) := \{s \in S : f(s) > t\}$$ and $$\mathbf{1}_{(t_{n+1} < f \leq t_n)}$$ denotes the indicator function of the set $$(t_{n+1} < f \leq t_n) := \{s \in S : t_{n+1} < f(s) \leq t_n\}.$$ The sequence $$(t_n)_{n \in \Z}$$ is decreasing and converges to $$0$$ as $$n \to \infty.$$ Consequently, if $$t_n = 0$$ then $$t_{n+1} = 0$$ and $$(t_{n+1} < f \leq t_n) = \varnothing$$ so that $$f_n = \frac{1}{r_n} \, f \,\mathbf{1}_{(t_{n+1} < f \leq t_n)}$$ is identically equal to $$0$$ (in particular, the division $$\tfrac{1}{r_n}$$ by $$r_n = 0$$ causes no issues).

The complementary cumulative distribution function $$t \in \Reals \mapsto \mu(|f| > t)$$ of $$|f| = f$$ that was used to define the $$t_n$$ also appears in the definition of the weak $$L^p$$-norm (given below) and can be used to express the $$p$$-norm $$\|\cdot\|_p$$ (for $$1 \leq p < \infty$$) of $$f \in L^p(S, \mu)$$ as the integral $$\|f\|_p^p ~=~ p \, \int_0^\infty t^{p-1} \mu(|f| > t) \, \mathrm{d} t \, ,$$ where the integration is with respect to the usual Lebesgue measure on $$(0, \infty).$$

Dual spaces
The dual space (the Banach space of all continuous linear functionals) of $$L^p(\mu)$$ for $$1 < p < \infty$$ has a natural isomorphism with $$L^q(\mu),$$ where $$q$$ is such that $$\tfrac{1}{p} + \tfrac{1}{q} = 1$$ (i.e. $$q = \tfrac{p}{p-1}$$). This isomorphism associates $$g \in L^q(\mu)$$ with the functional $$\kappa_p(g) \in L^p(\mu)^*$$ defined by $$f \mapsto \kappa_p(g)(f) = \int f g \, \mathrm{d}\mu$$ for every $$f \in L^p(\mu).$$

The fact that $$\kappa_p(g)$$ is well defined and continuous follows from Hölder's inequality. $$\kappa_p : L^q(\mu) \to L^p(\mu)^*$$ is a linear mapping which is an isometry by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon–Nikodym theorem, see ) that any $$G \in L^p(\mu)^*$$ can be expressed this way: i.e., that $$\kappa_p$$ is onto. Since $$\kappa_p$$ is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that $$L^q(\mu)$$ is the continuous dual space of $$L^p(\mu).$$

For $$1 < p < \infty,$$ the space $$L^p(\mu)$$ is reflexive. Let $$\kappa_p$$ be as above and let $$\kappa_q : L^p(\mu) \to L^q(\mu)^*$$ be the corresponding linear isometry. Consider the map from $$L^p(\mu)$$ to $$L^p(\mu)^{**},$$ obtained by composing $$\kappa_q$$ with the transpose (or adjoint) of the inverse of $$\kappa_p:$$

$$j_p : L^p(\mu) \mathrel{\overset{\kappa_q}{\longrightarrow}} L^q(\mu)^* \mathrel{\overset{\left(\kappa_p^{-1}\right)^*}{\longrightarrow}} L^p(\mu)^{**}$$

This map coincides with the canonical embedding $$J$$ of $$L^p(\mu)$$ into its bidual. Moreover, the map $$j_p$$ is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure $$\mu$$ on $$S$$ is sigma-finite, then the dual of $$L^1(\mu)$$ is isometrically isomorphic to $$L^\infty(\mu)$$ (more precisely, the map $$\kappa_1$$ corresponding to $$p = 1$$ is an isometry from $$L^\infty(\mu)$$ onto $$L^1(\mu)^*.$$

The dual of $$L^\infty(\mu)$$ is subtler. Elements of $$L^\infty(\mu)^*$$ can be identified with bounded signed finitely additive measures on $$S$$ that are absolutely continuous with respect to $$\mu.$$ See ba space for more details. If we assume the axiom of choice, this space is much bigger than $$L^1(\mu)$$ except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo–Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of $$\ell^\infty$$ is $$\ell^1.$$

Embeddings
Colloquially, if $$1 \leq p < q \leq \infty,$$ then $$L^p(S, \mu)$$ contains functions that are more locally singular, while elements of $$L^q(S, \mu)$$ can be more spread out. Consider the Lebesgue measure on the half line $$(0, \infty).$$ A continuous function in $$L^1$$ might blow up near $$0$$ but must decay sufficiently fast toward infinity. On the other hand, continuous functions in $$L^\infty$$ need not decay at all but no blow-up is allowed. The precise technical result is the following. Suppose that $$0 < p < q \leq \infty.$$ Then:

 * 1) $$L^q(S, \mu) \subseteq L^p(S, \mu)$$ if and only if $$S$$ does not contain sets of finite but arbitrarily large measure (any finite measure, for example).
 * 2) $$L^p(S, \mu) \subseteq L^q(S, \mu)$$ if and only if $$S$$ does not contain sets of non-zero but arbitrarily small measure (the counting measure, for example).

Neither condition holds for the real line with the Lebesgue measure while both conditions holds for the counting measure on any finite set. In both cases the embedding is continuous, in that the identity operator is a bounded linear map from $$L^q$$ to $$L^p$$ in the first case, and $$L^p$$ to $$L^q$$ in the second. (This is a consequence of the closed graph theorem and properties of $$L^p$$ spaces.) Indeed, if the domain $$S$$ has finite measure, one can make the following explicit calculation using Hölder's inequality $$\ \|\mathbf{1}f^p\|_1 \leq \|\mathbf{1}\|_{q/(q-p)} \|f^p\|_{q/p}$$ leading to $$\ \|f\|_p \leq \mu(S)^{1/p - 1/q} \|f\|_q .$$

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity $$I : L^q(S, \mu) \to L^p(S, \mu)$$ is precisely $$\|I\|_{q,p} = \mu(S)^{1/p - 1/q}$$ the case of equality being achieved exactly when $$f = 1$$ $$\mu$$-almost-everywhere.

Dense subspaces
Throughout this section we assume that $$1 \leq p < \infty.$$

Let $$(S, \Sigma, \mu)$$ be a measure space. An integrable simple function $$f$$ on $$S$$ is one of the form $$f = \sum_{j=1}^n a_j \mathbf{1}_{A_j}$$ where $$a_j$$ are scalars, $$A_j \in \Sigma$$ has finite measure and $${\mathbf 1}_{A_j}$$ is the indicator function of the set $$A_j,$$ for $$j = 1, \dots, n.$$ By construction of the integral, the vector space of integrable simple functions is dense in $$L^p(S, \Sigma, \mu).$$

More can be said when $$S$$ is a normal topological space and $$\Sigma$$ its Borel $\sigma$–algebra, i.e., the smallest 𝜎–algebra of subsets of $$S$$ containing the open sets.

Suppose $$V \subseteq S$$ is an open set with $$\mu(V) < \infty.$$ It can be proved that for every Borel set $$A \in \Sigma$$ contained in $$V,$$ and for every $$\varepsilon > 0,$$ there exist a closed set $$F$$ and an open set $$U$$ such that $$F \subseteq A \subseteq U \subseteq V \quad \text{and} \quad \mu(U) - \mu(F) = \mu(U \setminus F) < \varepsilon$$

It follows that there exists a continuous Urysohn function $$0 \leq \varphi \leq 1$$ on $$S$$ that is $$1$$ on $$F$$ and $$0$$ on $$S \setminus U,$$ with $$\int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon \, .$$

If $$S$$ can be covered by an increasing sequence $$(V_n)$$ of open sets that have finite measure, then the space of $$p$$–integrable continuous functions is dense in $$L^p(S, \Sigma, \mu).$$ More precisely, one can use bounded continuous functions that vanish outside one of the open sets $$V_n.$$

This applies in particular when $$S = \Reals^d$$ and when $$\mu$$ is the Lebesgue measure. The space of continuous and compactly supported functions is dense in $$L^p(\Reals^d).$$ Similarly, the space of integrable step functions is dense in $$L^p(\Reals^d);$$ this space is the linear span of indicator functions of bounded intervals when $$d = 1,$$ of bounded rectangles when $$d = 2$$ and more generally of products of bounded intervals.

Several properties of general functions in $$L^p(\Reals^d)$$ are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on $$L^p(\Reals^d),$$ in the following sense: $$\forall f \in L^p \left(\Reals^d\right) : \quad \left\|\tau_t f - f \right\|_p \to 0,\quad \text{as } \Reals^d \ni t \to 0,$$ where $$(\tau_t f)(x) = f(x - t).$$

Closed subspaces
If $$0 < p < \infty$$ is any positive real number, $$\mu$$ is a probability measure on a measurable space $$(S, \Sigma)$$ (so that $$L^\infty(\mu) \subseteq L^p(\mu)$$), and $$V \subseteq L^\infty(\mu)$$ is a vector subspace, then $$V$$ is a closed subspace of $$L^p(\mu)$$ if and only if $$V$$ is finite-dimensional ($$V$$ was chosen independent of $$p$$). In this theorem, which is due to Alexander Grothendieck, it is crucial that the vector space $$V$$ be a subset of $$L^\infty$$ since it is possible to construct an infinite-dimensional closed vector subspace of $$L^1\left(S^1, \tfrac{1}{2\pi}\lambda\right)$$ (that is even a subset of $$L^4$$), where $$\lambda$$ is Lebesgue measure on the unit circle $$S^1$$ and $$\tfrac{1}{2\pi} \lambda$$ is the probability measure that results from dividing it by its mass $$\lambda(S^1) = 2 \pi.$$

$ℓ$
Let $$(S, \Sigma, \mu)$$ be a measure space. If $$0 < p < 1,$$ then $$L^p(\mu)$$ can be defined as above: it is the quotient vector space of those measurable functions $$f$$ such that $$N_p(f) = \int_S |f|^p\, d\mu < \infty.$$

As before, we may introduce the $$p$$-norm $$\|f\|_p = N_p(f)^{1/p},$$ but $$\|\cdot\|_p$$ does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality $$(a + b)^p \leq a^p + b^p,$$ valid for $$a, b \geq 0,$$ implies that $$N_p(f + g) \leq N_p(f) + N_p(g)$$ and so the function $$d_p(f ,g) = N_p(f - g) = \|f - g\|_p^p$$ is a metric on $$L^p(\mu).$$ The resulting metric space is complete; the verification is similar to the familiar case when $$p \geq 1.$$ The balls $$B_r = \{f \in L^p : N_p(f) < r\}$$ form a local base at the origin for this topology, as $$r > 0$$ ranges over the positive reals. These balls satisfy $$B_r = r^{1/p} B_1$$ for all real $$r > 0,$$ which in particular shows that $$B_1$$ is a bounded neighborhood of the origin; in other words, this space is locally bounded, just like every normed space, despite $$\|\cdot\|_p$$ not being a norm.

In this setting $$L^p$$ satisfies a reverse Minkowski inequality, that is for $$u, v \in L^p$$ $$\Big\||u| + |v|\Big\|_p \geq \|u\|_p + \|v\|_p$$

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces $$L^p$$ for $$1 < p < \infty$$.

The space $$L^p$$ for $$0 < p < 1$$ is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in $$\ell^p$$ or $$L^p([0, 1]),$$ every open convex set containing the $$0$$ function is unbounded for the $$p$$-quasi-norm; therefore, the $$0$$ vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space $$S$$ contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in $$L^p([0, 1])$$ is the entire space. As a particular consequence, there are no nonzero continuous linear functionals on $$L^p([0, 1]);$$ the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space $$L^p(\mu) = \ell^p$$), the bounded linear functionals on $$\ell^p$$ are exactly those that are bounded on $$\ell^1,$$ namely those given by sequences in $$\ell^\infty.$$ Although $$\ell^p$$ does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on $$\Reals^n,$$ rather than work with $$L^p$$ for $$0 < p < 1,$$ it is common to work with the Hardy space $ℓ$ whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn–Banach theorem still fails in $L^{p}(μ) ⊂ L^{q}(μ)$ for $$p < 1$$.

$L^{p} (0 < p < 1)$, the space of measurable functions
The vector space of (equivalence classes of) measurable functions on $$(S, \Sigma, \mu)$$ is denoted $$L^0(S, \Sigma, \mu)$$. By definition, it contains all the $$L^p,$$ and is equipped with the topology of convergence in measure. When $$\mu$$ is a probability measure (i.e., $$\mu(S) = 1$$), this mode of convergence is named convergence in probability. The space $$L^0$$ is always a topological abelian group but is only a topological vector space if $$\mu(S)<\infty.$$ This is because scalar multiplication is continuous if and only if $$\mu(S)<\infty.$$ If $$(S,\Sigma,\mu)$$ is $$\sigma$$-finite then the weaker topology of local convergence in measure is an F-space, i.e. a completely metrizable topological vector space. Moreover, this topology is isometric to global convergence in measure $$(S,\Sigma,\nu)$$ for a suitable choice of probability measure $$\nu.$$

The description is easier when $$\mu$$ is finite. If $$\mu$$ is a finite measure on $$(S, \Sigma),$$ the $$0$$ function admits for the convergence in measure the following fundamental system of neighborhoods $$V_\varepsilon = \Bigl\{f : \mu \bigl(\{x : |f(x)| > \varepsilon\} \bigr) < \varepsilon \Bigr\}, \qquad \varepsilon > 0.$$

The topology can be defined by any metric $$d$$ of the form $$d(f, g) = \int_S \varphi \bigl(|f(x) - g(x)|\bigr)\, \mathrm{d}\mu(x)$$ where $$\varphi$$ is bounded continuous concave and non-decreasing on $$[0, \infty),$$ with $$\varphi(0) = 0$$ and $$\varphi(t) > 0$$ when $$t > 0$$ (for example, $$\varphi(t) = \min(t, 1).$$ Such a metric is called Lévy-metric for $$L^0.$$ Under this metric the space $$L^0$$ is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if $$\mu(S)<\infty$$. To see this, consider the Lebesgue measurable function $$f:\mathbb R\rightarrow \mathbb R$$ defined by $$f(x)=x$$. Then clearly $$\lim_{c\rightarrow 0}d(cf,0)=\infty$$. The space $$L^0$$ is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure $$\lambda$$ on $$\Reals^n,$$ the definition of the fundamental system of neighborhoods could be modified as follows $$W_\varepsilon = \left\{f : \lambda \left(\left\{x : |f(x)| > \varepsilon \text{ and } |x| < \tfrac{1}{\varepsilon}\right\}\right) < \varepsilon\right\}$$

The resulting space $$L^0(\Reals^n, \lambda)$$, with the topology of local convergence in measure, is isomorphic to the space $$L^0(\Reals^n, g \, \lambda),$$ for any positive $$\lambda$$–integrable density $$g.$$

Weak $H$
Let $$(S, \Sigma, \mu)$$ be a measure space, and $$f$$ a measurable function with real or complex values on $$S.$$ The distribution function of $$f$$ is defined for $$t \geq 0$$ by $$\lambda_f(t) = \mu\{x \in S : |f(x)| > t\}.$$

If $$f$$ is in $$L^p(S, \mu)$$ for some $$p$$ with $$1 \leq p < \infty,$$ then by Markov's inequality, $$\lambda_f(t) \leq \frac{\|f\|_p^p}{t^p}$$

A function $$f$$ is said to be in the space weak $$L^p(S, \mu)$$, or $$L^{p,w}(S, \mu),$$ if there is a constant $$C > 0$$ such that, for all $$t > 0,$$ $$\lambda_f(t) \leq \frac{C^p}{t^p}$$

The best constant $$C$$ for this inequality is the $$L^{p,w}$$-norm of $$f,$$ and is denoted by $$\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{1/p}(t).$$

The weak $$L^p$$ coincide with the Lorentz spaces $$L^{p,\infty},$$ so this notation is also used to denote them.

The $$L^{p,w}$$-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for $$f$$ in $$L^p(S, \mu),$$ $$\|f\|_{p,w} \leq \|f\|_p$$ and in particular $$L^p(S, \mu) \subset L^{p,w}(S, \mu).$$

In fact, one has $$\|f\|^p_{L^p} = \int |f(x)|^p d\mu(x) \geq \int_{\{|f(x)| > t \}} t^p + \int_{\{|f(x)| \leq t \}} |f|^p \geq t^p \mu(\{|f| > t \}),$$ and raising to power $$1/p$$ and taking the supremum in $$t$$ one has $$\|f\|_{L^p} \geq \sup_{t > 0} t \; \mu(\{|f| > t \})^{1/p} = \|f\|_{L^{p,w}}.$$

Under the convention that two functions are equal if they are equal $$\mu$$ almost everywhere, then the spaces $$L^{p,w}$$ are complete.

For any $$0 < r < p$$ the expression $$\|| f |\|_{L^{p,\infty}} = \sup_{0<\mu(E)<\infty} \mu(E)^{-1/r + 1/p} \left(\int_E |f|^r\, d\mu\right)^{1/r}$$ is comparable to the $$L^{p,w}$$-norm. Further in the case $$p > 1,$$ this expression defines a norm if $$r = 1.$$ Hence for $$p > 1$$ the weak $$L^p$$ spaces are Banach spaces.

A major result that uses the $$L^{p,w}$$-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.

Weighted $H$ spaces
As before, consider a measure space $$(S, \Sigma, \mu).$$ Let $$w : S \to [a, \infty), a > 0$$ be a measurable function. The $$w$$-weighted $$L^p$$ space is defined as $$L^p(S, w \, \mathrm{d} \mu),$$ where $$w \, \mathrm{d} \mu$$ means the measure $$\nu$$ defined by $$\nu(A) \equiv \int_A w(x) \, \mathrm{d} \mu (x), \qquad A \in \Sigma,$$

or, in terms of the Radon–Nikodym derivative, $$w = \tfrac{\mathrm{d} \nu}{\mathrm{d} \mu}$$ the norm for $$L^p(S, w \, \mathrm{d} \mu)$$ is explicitly $$\|u\|_{L^p(S, w \, \mathrm{d} \mu)} \equiv \left(\int_S w(x) |u(x)|^p \, \mathrm{d} \mu(x)\right)^{1/p}$$

As $$L^p$$-spaces, the weighted spaces have nothing special, since $$L^p(S, w \, \mathrm{d} \mu)$$ is equal to $$L^p(S, \mathrm{d} \nu).$$ But they are the natural framework for several results in harmonic analysis ; they appear for example in the Muckenhoupt theorem: for $$1 < p < \infty,$$ the classical Hilbert transform is defined on $$L^p(\mathbf{T}, \lambda)$$ where $$\mathbf{T}$$ denotes the unit circle and $$\lambda$$ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on $$L^p(\Reals^n, \lambda).$$ Muckenhoupt's theorem describes weights $$w$$ such that the Hilbert transform remains bounded on $$L^p(\mathbf{T}, w \, \mathrm{d} \lambda)$$ and the maximal operator on $$L^p(\Reals^n, w \, \mathrm{d} \lambda).$$

$L^{0}$ spaces on manifolds
One may also define spaces $$L^p(M)$$ on a manifold, called the intrinsic $$L^p$$ spaces of the manifold, using densities.

Vector-valued $L^{p}$ spaces
Given a measure space $$(\Omega, \Sigma, \mu)$$ and a locally convex space $$E$$ (here assumed to be complete), it is possible to define spaces of $$p$$-integrable $$E$$-valued functions on $$\Omega$$ in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and then endow them with locally convex TVS-topologies that are (each in their own way) a natural generalization of the usual $$L^p$$ topology. Another way involves topological tensor products of $$L^p(\Omega, \Sigma, \mu)$$ with $$E.$$ Element of the vector space $$L^p(\Omega, \Sigma, \mu) \otimes E$$ are finite sums of simple tensors $$f_1 \otimes e_1 + \cdots + f_n \otimes e_n,$$ where each simple tensor $$f \times e$$ may be identified with the function $$\Omega \to E$$ that sends $$x \mapsto e f(x).$$ This tensor product $$L^p(\Omega, \Sigma, \mu) \otimes E$$ is then endowed with a locally convex topology that turns it into a topological tensor product, the most common of which are the projective tensor product, denoted by $$L^p(\Omega, \Sigma, \mu) \otimes_\pi E,$$ and the injective tensor product, denoted by $$L^p(\Omega, \Sigma, \mu) \otimes_\varepsilon E.$$ In general, neither of these space are complete so their completions are constructed, which are respectively denoted by $$L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\pi E$$ and $$L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\varepsilon E$$ (this is analogous to how the space of scalar-valued simple functions on $$\Omega,$$ when seminormed by any $$\|\cdot\|_p,$$ is not complete so a completion is constructed which, after being quotiented by $$\ker \|\cdot\|_p,$$ is isometrically isomorphic to the Banach space $$L^p(\Omega, \mu)$$). Alexander Grothendieck showed that when $$E$$ is a nuclear space (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

title: Lp space


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Vizcaya

Vizcaya (Spanish, 'Biscay') may refer to:

Places

 * Biscay, a Basque region and a province of Spain
 * Biscay (Congress of Deputies constituency)
 * Biscay (Basque Parliament constituency)
 * Biscay (Senate constituency)
 * Vizcaya Museum and Gardens, formerly Villa Vizcaya, in Miami, Florida, U.S.
 * Vizcaya station, in Miami, Florida, U.S.

People

 * Pierre de Vizcaya (1894–1933), Spanish motor racer

Other uses

 * Vizcaya (planthopper), a genus of insects
 * Spanish cruiser Vizcaya, a Spanish Navy armored cruiser that fought in the Spanish–American War
 * Vizcaya (1890), Filipino World War II era freighter

title: Vizcaya