# Bochner space

In mathematics, Bochner spaces are a generalization of the concept of Lp 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 Lp(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 Lp space. Thus, if X is the set of complex numbers, it is the standard Lebesgue Lp space.

Almost all standard results on Lp spaces do hold on Bochner spaces too; in particular, the Bochner spaces Lp(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 Lp(TX) 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 Lp spaces, Lp(TX) 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 Lp(TX) 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.)