One of the most common

classes of

vector space encountered in

mathematics, especially

functional analysis.

**Definition:** A Banach space is a normed vector space which is complete in the metric induced by its norm.

That is, start with a vector space E over the real numbers **R** or complex numbers **C** (or you could use another complete normed field such as the p-adic numbers, but that's a whole 'nother can of worms). Call your field of choice **k**. Define a norm || || on E, which is a notion of absolute value or distance from zero that is compatible with the vector space structure on E:

If you think of ||x|| as the

distance of x from

zero, then || || defines a

metric d on E, by setting d(x, y) = ||x - y||. This turns E into a

metric space. If E is

complete under this

metric, then we call E a

Banach space.

Every finite-dimensional vector space over the real or complex numbers is a Banach space, using the conventional Euclidean norm

||(x_{1}, ..., x_{n})|| = (|x_{1}|^{2} + ... + |x_{n}|^{2})^{1/2}.

(A perverse way to describe this norm on **R**^{n} would be to call it L^{2}_{R}{1, ..., n}. See below.) In fact, it is not hard to prove that every finite-dimensional Banach space is isomorphic to one of these. The interest of Banach spaces lies in the many nontrivial infinite-dimensional examples. These include:

- C
_{R}(X) or C_{C}(X), the real- or complex-valued continuous functions on a compact metric space X, using the "uniform norm"
||f||_{∞} = sup {|f(x)|: x ∈ X}.

(Why the ∞ subscript? Because this is the same norm used for L^{∞}, in the next item.)

- L
^{p}(μ), the Lebesgue space of functions on a measure space (X, μ) whose pth powers are integrable; when 1 ≤ p < ∞, the L^{p} norm of a function f is the pth root of the integral of |f|^{p} over X:
||f||_{p} = (∫_{X} |f|^{p} dμ)^{1/p}.

For p = ∞, the corresponding space L^{∞}(μ) is the space of measurable functions f on X which are *essentially bounded*, that is there exists some bound N such that {x ∈ X : |f(x)| > N} has μ-measure zero. The infimum of all such N is called the *essential supremum* of |f| on X, and we put

||f||_{∞} = ess sup_{x∈X} |f(x)|.

(Why is this the *same* norm as above? Because a *continuous* function, which is essentially bounded, is bounded.) When the total measure μ(X) is finite, L^{∞} is actually the intersection of L^{p} for all 1 ≤ p < ∞; but this is no longer true when X has infinite total measure.

Because Lebesgue integration ignores sets of measure zero, the elements of L^{p} are actually not functions, but equivalence classes of functions which are equal almost everywhere. The fundamental fact of Fourier analysis is an isometric isomorphism between L^{2}(**T**) and L^{2}(**Z**), where **Z** is the integers and **T** is the unit circle. (The latter space is frequently written *l*^{2}, but with a script lowercase 'ell'.) Elements of these two spaces are periodic functions and Fourier series respectively.

- C
^{k}(X), the space of k times differentiable functions on a compact smooth manifold X. Although, as sets of functions, C^{k}(X) is a subset of C(X), it is not a closed subspace of the Banach space C(X), because a uniformly convergent sequence of differentiable functions need only be continuous, not differentiable. To make C^{k}(X) into a Banach space we therefore need to restrict the notion of convergence by changing the norm. Roughly speaking, the C^{k} norm of a C^{k} function f on X is the sum of the uniform norms of f and its first k derivatives; there are some technical details involved in computing the norms of the derivatives, since they are not numbers or matrices but symmetric tensor fields on X. Unfortunately C^{∞}(X), the space of infinitely differentiable functions on X, is not a Banach space, but only a Fréchet space. Also, if X is not compact (as is the case for open subsets of **R**^{n}, for instance) these spaces are more complicated, which accounts for some of the technical difficulties in distribution theory.
- Many more technical examples, including the Sobolev spaces near and dear to my heart.