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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of, formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be useful for the study of differential and integral equations; the usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had been introduced in 1887 by the Italian mathematician and physicist Vito Volterra; the theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy.

Hadamard founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach. In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals with finite-dimensional spaces, does not use topology. An important part of functional analysis is the extension of the theory of measure and probability to infinite dimensional spaces known as infinite dimensional analysis; the basic and first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space; these spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics. More functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead to the definition of C*-algebras and other operator algebras. Hilbert spaces can be classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are understood in linear algebra, infinite-dimensional separable Hilbert spaces are isomorphic to ℓ 2. Separability being important for applications, functional analysis of Hilbert spaces mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have been proven. General Banach spaces are more complicated than Hilbert spaces, cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis. Examples of Banach spaces are L p -spaces for any real number p ≥ 1.

Given a measure μ on set X L p, sometimes denoted L p or L p, has as its vectors equivalence classes of measurable functions whose absolute value's p -th power has finite integral, that is, functions f for which one has ∫ X | f | p d μ < + ∞. If μ is the counting measure the integral may be replaced by a sum; that is, we require ∑ x ∈ X | f | p < + ∞. It is not necessary to deal with equivalence classes, the space is denoted ℓ p, written more ℓ p in the case when X is the set of non-negative integers. In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, the dual of its dual space; the corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not be isometrically isomorphic in any way, contrary to the finite-dimensional situation; this is explained in the dual space article.

The notion of derivative can be extended to arbitrary functions be

East End Historic District (Valdosta, Georgia)

The East End Historic District in Valdosta, Georgia is a 255 acres historic district, listed on the National Register of Historic Places in 2005. The district is northeast of downtown Valdosta and is bounded by North Ashley and E. Ann Sts. East Hill Ave. and the Georgia and Florida Railroad tracks. The district included 470 contributing buildings, a contributing structure, a contributing site, it includes Smith Park. It includes work by local architect Lloyd Greer and it includes Queen Anne and Colonial Revival architecture. Media related to East End Historic District at Wikimedia Commons

Fading Frontier

Fading Frontier is the seventh studio album by the American indie rock band Deerhunter, released on October 16, 2015 on 4AD. Produced Ben H. Allen, who had worked with the band on Halcyon Digest, the band itself, the album was preceded by the singles "Snakeskin", "Breaker" and "Living My Life". In December 2014, Bradford Cox was involved in a car crash which left him "seriously injured, but provided a perspective-giving jolt". Prior to the release of Fading Frontier, Cox stated that the accident "erased all illusions" and admitted that it was a definite turning point for him. Fading Frontier is Cox's first release since the accident; this album saw the departure of Frankie Broyles, who left the band in 2015 to focus on his solo career. The song "Ad Astra" contains a sample from Bascom Lamar Lunsford's "I Wish I Was a Mole in the Ground." Fading Frontier was announced via a countdown timer on Deerhunter's website, which ended on 16 August 2015. The first single released was "Snakeskin" on August 17, followed up with "Breaker" on September 15, of which the video was directed by Cox himself.

The cover art for the album is a photograph titled "Zuma" by the artist John Divola. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream critics, the album has received an average score of 81, based on 27 reviews, indicating "universal acclaim."Consequence of Sound critic Sheldon Pearce wrote that the album is an "one more gem from a well-traveled band that’s still finding new territory to explore." NME critic Barry Nicholson praised the record and gave it a perfect score, describing it as "a remarkable album, one that only grows more awesome with each listen." Ian Cohen of Pitchfork awarded the album with a "Best New Music" tag and wrote: "If there isn't a Deerhunter sound, there's a Deerhunter perspective that runs through their work, best summed up in'All the Same'—'take your handicaps/ Channel them and feed them back/ Until they become your strengths.' The weird era continues." In The Guardian, Alex Petridis noted the album's more mainstream sound: "there are so many straightforwardly commercial-sounding songs here," he observed, "that Fading Frontier could conceivably be an album that turns Deerhunter from cult concern into mainstream success."

Citing a lack of angst and urgency in comparison to previous efforts, Exclaim!'s Anna Alger wrote that "On Fading Frontier, Deerhunter focus on their ability as a band to hypnotize and confound, which make the explosive moments here stand out that much more."In a less favorable review, Clash critic Sam Walker-Smart wrote that the record "is by no means a poor album, truth be told doesn't possess a bad number on it." He added: "When the oddities on this album ride so high they should have let complete weirdness take over." All tracks are written except "Ad Astra" by Lockett Pundt. Credits for the album were adapted from a press release by 4AD. Bradford Cox – lead vocals, percussion, electronics Lockett Pundt – guitar, co-lead vocals, lead vocals, keyboards Moses Archuletadrums, electronics Josh McKaybass guitar, organ Tim Gane – electronic harpsichord James Cargillsynthesizers, tapes Zumi Rosow – treated alto saxophone "Deerhunter" on 4AD Records