BANACH SPACE

  

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BANACH SPACE

Specialty Definition: BANACH SPACE

DomainDefinition

Computing

Banach space A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. All finite-dimensional real and complex normed vector spaces are complete and thus are are Banach spaces. Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals. Several theorems hold only in Banach spaces, e.g. the Banach inverse mapping theorem. All finite-dimensional real and complex vector spaces are Banach spaces. Hilbert spaces, spaces of integrable functions, and spaces of absolutely convergent series are examples of infinite-dimensional Banach spaces. Applications include wavelets, signal processing, and radar. [Robert E. Megginson, "An Introduction to Banach Space Theory", Graduate Texts in Mathematics, 183, Springer Verlag, September 1998]. (2000-03-10). Source: The Free On-line Dictionary of Computing.

Source: compiled by the editor from various references; see credits.

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Specialty Definition: Banach space

(From Wikipedia, the free Encyclopedia)

In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions.

Definition

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||.|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||x - y||) in V has a limit in V.

Examples

Throughout, let K stand for one of the fields R or C.

The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, ..., xn) is given by ||x|| = (∑ |xi|2)1/2, is a Banach space.

The space of all continuous functions f : [a, b] -> K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a function as ||f|| = sup { |f(x)| : x in [a, b] }. This is indeed a norm since continuous functions defined on a closed interval are bounded. The space is complete under this norm, and the resulting Banach space is denoted by C[a, b]. This example can be generalized to the space C(X) of all continuous functions X -> K, where X is a compact space, or to the space of all bounded continuous functions X -> K, where X is any topological space, or indeed to the space B(X) of all bounded functions X -> K, where X is any set. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact unitary Banach algebras.

If p ≥ 1 is a real number, we can consider the space of all infinite sequences (x1, x2, x3, ...) of elements in K such that the infinite series ∑ |xi|p converges. The p-th root of this series' value is then defined to be the p-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by l p.

The Banach space l∞ consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members.

Again, if p ≥ 1 is a real number, we can consider all functions f : [a, b] -> K such that |f|p is Lebesgue integrable. The p-th root of this integral is then defined to be the norm of f. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: f and g are equivalent if and only if the norm of f - g is zero. The set of equivalence classes then forms a Banach space; it is denoted by L p[a, b]. It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see L p spaces for details.

Finally, every Hilbert space is a Banach space. The converse is not true.

Linear operators

If V and W are Banach spaces over the same ground field K, the set of all continuous K-linear maps A : V -> W is denoted by L(V, W). Note that in infinite-dimensional spaces, not all linear maps are automatically continuous. L(V, W) is a vector space, and by defining the norm ||A|| = sup { ||Ax|| : x in V with ||x|| ≤ 1 } it can be turned into a Banach space.

The space L(V) = L(V, V) even forms a unitary Banach algebra; the multiplication operation is given by the composition of linear maps.

Derivatives

It is possible to define the derivative of a function f : V -> W between two Banach spaces. Intuitively, if x is an element of V, the derivative of f at the point x is a continuous linear map which approximates f near x.

Formally, f is called differentiable at x if there exists a continuous linear map A : V -> W such that

limh->0 ||f(x + h) - f(x) - A(h)|| / ||h||    =     0
The limit here is taken over all sequences of non-zero elements in V which converge to 0. If the limit exists, we write Df(x) = A and call it the derivative of f at x.

This notion of derivative is in fact a generalization of the ordinary derivative of functions R -> R, since the linear maps from R to R are just multiplications with real numbers.

If f is differentiable at every point x of V, then Df : V -> L(V, W) is another map between Banach spaces (in general not a linear map!), and can possibly be differentiated again, thus defining the higher derivatives of f. The n-th derivative at a point x can then be viewed as a multilinear map Vn -> W.

Differentiation is a linear operation in the following sense: if f and g are two maps V - W which are differentiable at x, and r and s are scalars from K, then rf + sg is differentiable at x with D(rf + sg)(x) = rD(f)(x) + sD(g)(x).

The chain rule is also valid in this context: if f : V -> W is differentiable at x in V, and g : W -> X is differentiable in f(x), then the composition g o f is differentiable in x and the derivative is the composition of the derivatives:

D(g o f)(x) = D(g\)(f(x)) o D(f)(x)

Dual space

If V is a Banach space and K is the underlying field (either the real or the complex numbers), then K is itself a Banach space (using the absolute value as norm) and we can define the dual space V' by V' = L(V, K). This is again a Banach space. It can be used to define a new topology on V: the weak topology.

There is a natural map F from V to V'' defined by

F(x)(f) = f(x)
for all x in V and f in V'. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R -> R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Banach space."

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Crosswords: BANACH SPACE

Specialty definitions using "BANACH SPACE": Banach algebra, Banach inverse mapping theorem. (references)

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Commercial Usage: BANACH SPACE

DomainTitle

Books

  • Summing and Nuclear Norms in Banach Space Theory (reference)

  • Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (Mathematical Surveys and Monographs, 49) (reference)

  • Orthonormal Systems and Banach Space Geometry (Encyclopedia of Mathematics and Its Applications, 70) (reference)

  • Spectral Theory of Banach Space Operators: Ck Superscript-Classification, Abstract Volterra Operators, Similarity, Spectrality, Local Spectral Analysis (Lecture Notes in Mathematics (Springer-Verlag), 1012.) (reference)

    (more book examples)

Source: compiled by the editor from various references; see credits.

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Frequency of Internet Keywords: BANACH SPACE

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.
 
ExpressionFrequency
per Day

banach space

3
Source: compiled by the editor from various references; see credits.

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Modern Translation: BANACH SPACE

Language Translations for "BANACH SPACE"; alternative meanings/domain in parentheses.

German

  

Banachraum. (various references)

   

Pig Latin

  

anachbay acespay

Source: compiled by the editor from various translation references.

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Anagrams: BANACH SPACE

Scrabble® Enable2K-Verified Anagrams

Words within the letters "a-a-a-b-c-c-e-h-n-p-s"

-3 letters: anaphase, panaceas, panaches.

-4 letters: apaches, cabanas, canapes, canchas, chances, panacea, panache, pechans, saphena.

-5 letters: abacas, anabas, apache, apneas, ashcan, baccae, baches, becaps, cabana, caches, canape, cancha, casaba, casbah, chance, chapes, cheaps, encash, hances, nachas, naches, pachas, paeans, paesan, pecans, pechan, shapen.

Source: compiled by the editor from various references; see credits.

SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

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Alternative Orthography: BANACH SPACE


Hexadecimal (or equivalents, 770AD-1900s) (references)

42 41 4E 41 43 48      53 50 41 43 45

Leonardo da Vinci (1452-1519; backwards) (references)

    

Binary Code (1918-1938, probably earlier) (references)

01000010 01000001 01001110 01000001 01000011 01001000 00100000 01010011 01010000 01000001 01000011 01000101

HTML Code (1990) (references)

&#66 &#65 &#78 &#65 &#67 &#72 &#32 &#83 &#80 &#65 &#67 &#69

ISO 10646 (1991-1993) (references)

0042 0041 004E 0041 0043 0048      0053 0050 0041 0043 0045

Encryption (beginner's substitution cypher): (references)

36354835374225350353739

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INDEX

1. Crosswords
2. Usage: Commercial
3. Expressions: Internet
4. Translations: Modern 		5. Anagrams
6. Orthography
7. Bibliography


  

Copyright © Philip M. Parker, INSEAD. Terms of Use.