Bochner space

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In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to more general domains and ranges than the initial definition, specifically by replacing the Lebesgue integral with the Bochner integral. 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, Σ, μ), a Banach space (X, || · ||_X) and 1 ≤ p ≤ +∞, the Bochner space L^p(T; X) is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all 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 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 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 L^p(T; X) rather than an equivalence class (which would be more technically correct).

[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 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 (or, equivalently, have compact support in Ω); $H - 1 (Ω)$ denotes the dual space of $H_{0}^{1} (\Omega)$ .