Bochner space
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
which is not necessarily the space \R or \Complex 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 Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
\, 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. Bochner spaces are named for the
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Aust ...
.


Definition

Given a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
(T, \Sigma; \mu), a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
\left(X, \, \,\cdot\,\, _X\right) and 1 \leq p \leq \infty, the Bochner space L^p(T; X) is defined to be the
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing t ...
(by equality
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
) of the space of all Bochner measurable functions u : T \to X such that the corresponding norm is finite: \, u\, _ := \left( \int_ \, u(t) \, _^ \, \mathrm \mu (t) \right)^ < + \infty \mbox 1 \leq p < \infty, \, u\, _ := \mathrm_ \, u(t)\, _ < + \infty. In other words, as is usual in the study of L^p spaces, L^p(T; X) is a space of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a \mu-
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null s ...
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).


Applications

Bochner spaces are often used in the
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
approach to the study of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s that depend on time, e.g. the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
: 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.


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 \mu will be one-dimensional
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. 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 \Omega in \R^n and an interval of time
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
one seeks solutions u \in L^2\left(
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
H_0^1(\Omega)\right) with time derivative \frac \in L^2 \left(
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
H^(\Omega)\right). Here H_0^1(\Omega) denotes the Sobolev
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of once- weakly differentiable functions with first weak derivative in L^2(\Omega) that vanish at the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
support in Ω); H^ (\Omega) denotes the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of H_0^1(\Omega). (The "
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
" with respect to time t above is actually a
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with resp ...
, since the use of Bochner spaces removes the space-dependence.)


See also

* * * * *


References

* {{Functional analysis Functional analysis Partial differential equations Sobolev spaces Lp spaces