Bochner space

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In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to slightly more general domains and ranges than the initial definition. They are often used in the functional analysis approach to the study of partial differential equations that depend on time, e.g. the heat equation.

Bochner spaces are named for the Polish-American mathematician Salomon Bochner.

[edit] Definition

Given a measure space $(T, \mathcal{F}, \mu)$ , a Banach space $(X, \| - \|_{X})$ and $1 \leq p \leq + \infty$ , the Bochner space $L p (T; X)$ is defined to be the space of all measurable functions $u : T \to X$ such that the corresponding norm is finite:

$\| u \|_{L^{p} (T; X)} := \left( \int_{T} \| u(t) \|_{X}^{p} \, \mathrm{d} \mu (t) \right)^{1/p} < + \infty$ for $1 \leq p < \infty,$

$\| u \|_{L^{\infty} (T; X)} := \mathrm{ess\,sup}_{t \in T} \| u(t) \|_{X} < + \infty.$

[edit] 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 $\Omega \subseteq \mathbb{R}^{n}$ 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 $L 2 (Ω)$ that vanish at the boundary of $Ω$ (or, equivalently, have compact support in $Ω$ ); $H - 1 (Ω)$ denotes the dual space of $H_{0}^{1} (\Omega)$ .

(The "partial derivative" with respect to time $t$ above is actually a full derivative, since the use of Bochner spaces removes the space-dependence.)