Bochner space

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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(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 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(X) are Banach spaces for 1\le p\le \infty.

Background[edit]

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

Applications[edit]

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

Definition[edit]

Given a measure space (T, Σ, μ), a Banach space (X, || · ||X) and 1 ≤ p ≤ +∞, the Bochner space Lp(TX) 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:

\| 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 Lp spaces, Lp(TX) 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(TX) rather than an equivalence class (which would be more technically correct).

Application to PDE theory[edit]

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 (in the sense of trace, or, equivalently, are limits of smooth functions with compact support in Ω); H^{-1} (\Omega) denotes the dual space of H_{0}^{1} (\Omega).

(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[edit]

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

See also[edit]