# 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||*lies in the standard

_{X}*L*space. Thus, if

^{p}*X*is the set of complex numbers, it is the standard Lebesgue

*L*space.

^{p}Almost all standard results on *L ^{p}* spaces do hold on Bochner spaces too; in particular, the Bochner spaces

*L*are Banach spaces for .

^{p}(X)## 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 *L*²(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.)

## References

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