Bochner space


In mathematics, Bochner spaces are a generalization of the concept of Lp 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 Lp consists of all Bochner measurable functions f with values in the Banach space X whose norm ||f||X lies in the standard Lp space. Thus, if X is the set of complex numbers, it is the standard Lebesgue Lp space.
Almost all standard results on Lp spaces do hold on Bochner spaces too; in particular, the Bochner spaces Lp are Banach spaces for.

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 of functions of space, possibly in some Bochner space.

Definition

Given a measure space, a Banach space and 1 ≤ p ≤ +∞, the Bochner space Lp is defined to be the Kolmogorov quotient 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 Lp spaces, Lp 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 Lp rather than an equivalence class.

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 , one seeks solutions
with time derivative
Here denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in L² that vanish at the boundary of Ω ; denotes the dual space of.