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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.

Contents

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 ≤ p ≤ ∞ .

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 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

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

∥ u ∥ L p ( T ; X ) := ( ∫ T ∥ u ( t ) ∥ X p d μ ( t ) ) 1 / p < + ∞  for  1 ≤ p < ∞ , ∥ u ∥ L ∞ ( T ; X ) := e s s s u p t ∈ T ∥ u ( t ) ∥ X < + ∞ .

In other words, as is usual in the study of Lp spaces, Lp(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 Lp(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 Rn and an interval of time [0, T], one seeks solutions

u ∈ L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) )

with time derivative

∂ u ∂ t ∈ L 2 ( [ 0 , T ] ; H − 1 ( Ω ) ) .

Here H 0 1 ( Ω ) 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 ( Ω ) denotes the dual space of H 0 1 ( Ω ) .

(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

Bochner space Wikipedia
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