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 ${\displaystyle 1\leq p\leq \infty }$.

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 ${\displaystyle g(t,x)}$ is a scalar function of time and space, one can write ${\displaystyle (f(t))(x):=g(t,x)}$ 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:

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

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

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

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

with time derivative

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

Here ${\displaystyle H_{0}^{1}(\Omega )}$ 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); ${\displaystyle H^{-1}(\Omega )}$ denotes the dual space of ${\displaystyle H_{0}^{1}(\Omega )}$.

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

See also

• Vector-valued functions

References

• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.