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 $$\Complex$$ 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 $$1 \leq p \leq \infty.$$

Bochner spaces are named for the mathematician Salomon Bochner.

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 $$\R^n$$ 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.)