
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.
Throughout, let K stand for one of the fields R or C.
The familiar Euclidean spaces K^{n}, where the Euclidean norm of x = (x_{1}, ..., x_{n}) is given by x = (∑ x_{i}^{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 (x_{1}, x_{2}, x_{3}, ...) of elements in K such that the infinite series ∑ x_{i}^{p} converges. The pth root of this series' value is then defined to be the pnorm 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 pth 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 nonzero 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.
If V and W are Banach spaces over the same ground field K, the set of all continuous Klinear maps A : V > W is denoted by L(V, W). Note that in infinitedimensional 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.
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
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 nth derivative at a point x can then be viewed as a multilinear map V^{n} > 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:
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
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 LFspaces[?] are complete uniform vector spaces arising as limits of Fréchet spaces.
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