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||_X lies in the standard L^p space. Thus, if X is the set of complex numbers, it is the standard Lebesgue L^p space.

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

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 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:

\|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 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 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

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

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 3/26/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.