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 C 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 .
Contents
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 is a scalar function of time and space, one can write 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 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 → X such that the corresponding norm is finite:
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).
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 R^{n} and an interval of time [0, T], one seeks solutions
with time derivative
Here 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); denotes the dual space of .
(The "partial derivative" with respect to time t above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)
See also
References
- Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.