Banach space (idea)

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:

• ||x|| is a nonnegative real number for all x ∈ E, and ||x|| = 0 if and only if x = 0.
• if α ∈ k (α is a number) and x ∈ E, then ||αx|| = |α| ||x||, where |α| is measured using the usual absolute value on k.
• if x, y in E, then ||x + y|| ≤ ||x|| + ||y||. This is the familiar triangle inequality for absolute value.

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

||(x₁, ..., x_n)|| = (|x₁|² + ... + |x_n|²)^1/2.

(A perverse way to describe this norm on Rⁿ would be to call it L²_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 L²(T) and L²(Z), where Z is the integers and T is the unit circle. (The latter space is frequently written l², 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 Rⁿ, 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.