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# Banach space

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

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.