Bochner space

In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a Banach space which is not necessarily the space R or C of real or complex numbers.

The space L^p(X) consists of (equivalence classes of) all Bochner measurable functions f with values in the Banach space X whose norm ||f||_X lies in the standard L^p space. Thus, if X is the set of complex numbers, it is the standard Lebesgue L^p space.

Almost all standard results on L^p spaces do hold on Bochner spaces too; in particular, the Bochner spaces L^p(X) are Banach spaces for $1\leq p\leq \infty$ .

Background

Bochner spaces are named for the Polish-American mathematician Salomon Bochner.

Applications

Bochner spaces are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation: if the temperature $g(t,x)$ is a scalar function of time and space, one can write $(f(t))(x):=g(t,x)$ to make f a family f(t) (parametrized by time) of functions of space, possibly in some Bochner space.

Definition

Given a measure space (T, Σ, μ), a Banach space (X, || · ||_X) and 1 ≤ p ≤ +∞, the Bochner space L^p(T; X) is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions u : T → X such that the corresponding norm is finite:

\|u\|_{{L^{{p}}(T;X)}}:=\left(\int _{{T}}\|u(t)\|_{{X}}^{{p}}\,{\mathrm {d}}\mu (t)\right)^{{1/p}}<+\infty {\mbox{ for }}1\leq p<\infty ,

\|u\|_{{L^{{\infty }}(T;X)}}:={\mathrm {ess\,sup}}_{{t\in T}}\|u(t)\|_{{X}}<+\infty .

In other words, as is usual in the study of L^p spaces, L^p(T; X) is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a μ-measure zero subset of T. As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in L^p(T; X) rather than an equivalence class (which would be more technically correct).

Application to PDE theory

Very often, the space T is an interval of time over which we wish to solve some partial differential equation, and μ will be one-dimensional Lebesgue measure. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region Ω in Rⁿ and an interval of time [0, T], one seeks solutions

u\in L^{{2}}\left([0,T];H_{{0}}^{{1}}(\Omega )\right)

with time derivative

{\frac {\partial u}{\partial t}}\in L^{{2}}\left([0,T];H^{{-1}}(\Omega )\right).

Here $H_{{0}}^{{1}}(\Omega )$ denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in L²(Ω) that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); $H^{{-1}}(\Omega )$ denotes the dual space of $H_{{0}}^{{1}}(\Omega )$ .

(The "partial derivative" with respect to time t above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)

References