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 or of real or complex numbers.

The space consists of (equivalence classes of) all Bochner measurable functions with values in the Banach space whose norm lies in the standard space. Thus, if is the set of complex numbers, it is the standard Lebesgue space.

Almost all standard results on spaces do hold on Bochner spaces too; in particular, the Bochner spaces are Banach spaces for

Bochner spaces are named for the mathematician Salomon Bochner.


Given a measure space a Banach space and the Bochner space 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 spaces, is a space of equivalence classes of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a -measure zero subset of 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 to make a family (parametrized by time) of functions of space, possibly in some Bochner space.

Application to PDE theory

Very often, the 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 in and an interval of time one seeks solutions

with time derivative
Here 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.)

See also