Bochner space
Encyclopedia
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
.
-American
mathematician
Salomon Bochner
.
approach to the study of partial differential equation
s that depend on time, e.g. the heat equation
: if the temperature
is a scalar function of time and space, one can write 
to make f a function of time, with f(t) being a function of space, possibly in some Bochner 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 measurable function
s u : T → X such that the corresponding norm is finite:
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).
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
with time derivative
Here
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);
denotes the dual space
of
.
(The "partial derivative
" with respect to time t above is actually a total derivative
, since the use of Bochner spaces removes the space-dependence.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Bochner spaces are a generalization of the concept of Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
to functions whose values lie in a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
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
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
||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
.Background
Bochner spaces are named for the PolishPoland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
-American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Salomon Bochner
Salomon Bochner
Salomon Bochner was an American mathematician of Austrian-Hungarian origin, known for wide-ranging work in mathematical analysis, probability theory and differential geometry.- Life :...
.
Applications
Bochner spaces are often used in the functional analysisFunctional 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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
approach to the study of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s that depend on time, e.g. the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
: if the temperature
is a scalar function of time and space, one can write to make f a function of time, with f(t) being a function of space, possibly in some Bochner space.Definition
Given a measure space (T, Σ, μ), a Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
(X, || · ||X) and 1 ≤ p ≤ +∞, the Bochner space Lp(T; X) is defined to be the Kolmogorov quotient (by equality almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
) of the space of all measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s u : T → X such that the corresponding norm is finite:
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
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
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 intervalInterval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
of time over which we wish to solve some partial differential equation, and μ will be one-dimensional Lebesgue measure
Lebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
. 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
with time derivative
Here
denotes the SobolevSobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of once-weakly differentiable
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...
functions with first weak derivative in L²(Ω) that vanish at the boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
in Ω);
denotes the dual spaceDual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
of
.(The "partial derivative
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...
" with respect to time t above is actually a total derivative
Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...
, since the use of Bochner spaces removes the space-dependence.)