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 $1\le p\le \infty$.

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.)

See also

• Vector-valued functions

References

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