In mathematics, Bochner spaces are a generalization of the concept of
spaces to functions whose values lie in a Banach space which is not necessarily the space
of real or complex numbers.
consists of (equivalence classes of) all Bochner measurable functions
with values in the Banach space
lies in the standard
is the set of complex numbers, it is the standard Lebesgue
Almost all standard results on
Bochner spaces are named for the mathematician Salomon Bochner.
is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable functions
such that the corresponding norm is finite:
In other words, as is usual in the study of
is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a
As is also usual in the study of such spaces, it is usual to abuse notation and speak of a "function" in
rather than an equivalence class (which would be more technically correct).
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
is a scalar function of time and space, one can write
(parametrized by time) of functions of space, possibly in some Bochner space.
is an interval of time over which we wish to solve some partial differential equation, and
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
and an interval of time
one seeks solutions
with time derivative
denotes the Sobolev Hilbert space of once-weakly differentiable functions with first weak derivative in
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
above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)