Bochner space

In mathematics, Bochner spaces are a generalization of the concept of ${\displaystyle L^{p}}$ spaces to functions whose values lie in a Banach space which is not necessarily the space ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ of real or complex numbers.

The space ${\displaystyle L^{p}(X)}$ consists of (equivalence classes of) all Bochner measurable functions ${\displaystyle f}$ with values in the Banach space ${\displaystyle X}$ whose norm ${\displaystyle \|f\|_{X}}$ lies in the standard ${\displaystyle L^{p}}$ space. Thus, if ${\displaystyle X}$ is the set of complex numbers, it is the standard Lebesgue ${\displaystyle L^{p}}$ space.

Almost all standard results on ${\displaystyle L^{p}}$ spaces do hold on Bochner spaces too; in particular, the Bochner spaces ${\displaystyle L^{p}(X)}$ are Banach spaces for ${\displaystyle 1\leq p\leq \infty .}$

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

Definition[]

Given a measure space ${\displaystyle (T,\Sigma ;\mu ),}$ a Banach space ${\displaystyle \left(X,\|\,\cdot \,\|_{X}\right)}$ and ${\displaystyle 1\leq p\leq \infty ,}$ the Bochner space ${\displaystyle L^{p}(T;X)}$ is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions ${\displaystyle u:T\to X}$ such that the corresponding norm is finite:

In other words, as is usual in the study of ${\displaystyle L^{p}}$ spaces, ${\displaystyle L^{p}(T;X)}$ is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a ${\displaystyle \mu }$-measure zero subset of ${\displaystyle T.}$ As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in ${\displaystyle L^{p}(T;X)}$ rather than an equivalence class (which would be more technically correct).

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 ${\displaystyle g(t,x)}$ is a scalar function of time and space, one can write ${\displaystyle (f(t))(x):=g(t,x)}$ to make ${\displaystyle f}$ a family ${\displaystyle f(t)}$ (parametrized by time) of functions of space, possibly in some Bochner space.

Application to PDE theory[]

Very often, the space ${\displaystyle T}$ is an interval of time over which we wish to solve some partial differential equation, and ${\displaystyle \mu }$ 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 ${\displaystyle \Omega }$ in ${\displaystyle \mathbb {R} ^{n}}$ and an interval of time ${\displaystyle [0,T],}$ one seeks solutions

with time derivative Here ${\displaystyle H_{0}^{1}(\Omega )}$ denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in ${\displaystyle L^{2}(\Omega )}$ that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); ${\displaystyle H^{-1}(\Omega )}$ denotes the dual space of ${\displaystyle H_{0}^{1}(\Omega ).}$

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

See also[]

• Bochner integral
• Bochner measurable function
• Vector measure
• Vector-valued functions
• Weakly measurable function

References[]

• Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.