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Banach space

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.

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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.



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