##
Definition

Given a measure space $(T,\Sigma ;\mu ),$ a Banach space $\left(X,\|\,\cdot \,\|_{X}\right)$ and $1\leq p\leq \infty ,$ 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 $u:T\to X$ 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 $\mu$-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).

##
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.

###
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 $\mu$ 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 $\Omega$ in $\mathbb {R} ^{n))$ 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 $L^{2}(\Omega )$ that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); $H^{-1}(\Omega )$ 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.)