# Bochner space

Short description: Mathematical concept

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

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

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

Bochner spaces are named for the mathematician Salomon Bochner.

## Definition

Given a measure space $\displaystyle{ (T, \Sigma; \mu), }$ a Banach space $\displaystyle{ \left(X, \|\,\cdot\,\|_X\right) }$ and $\displaystyle{ 1 \leq p \leq \infty, }$ the Bochner space $\displaystyle{ L^p(T; X) }$ is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions $\displaystyle{ u : T \to X }$ such that the corresponding norm is finite: $\displaystyle{ \|u\|_{L^p(T; X)} := \left( \int_{T} \| u(t) \|_{X}^{p} \, \mathrm{d} \mu (t) \right)^{1/p} \lt + \infty \mbox{ for } 1 \leq p \lt \infty, }$ $\displaystyle{ \|u\|_{L^{\infty}(T; X)} := \mathrm{ess\,sup}_{t \in T} \|u(t)\|_{X} \lt + \infty. }$

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

## 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 $\displaystyle{ f }$ a family $\displaystyle{ f(t) }$ (parametrized by time) of functions of space, possibly in some Bochner space.

### Application to PDE theory

Very often, the space $\displaystyle{ T }$ is an interval of time over which we wish to solve some partial differential equation, and $\displaystyle{ \mu }$ 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 $\displaystyle{ \Omega }$ in $\displaystyle{ \R^n }$ and an interval of time $\displaystyle{ [0, T], }$ 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 $\displaystyle{ L^2(\Omega) }$ 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 $\displaystyle{ t }$ above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)

## References

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