In mathematics, Bochner spaces are a generalization of the concept of L^p spaces to functions whose values lie in a Banach space of real or complex numbers.

consists of (equivalence classes of) all Bochner measurable whose norm space.

Bochner spaces are named for the mathematician Salomon Bochner.

Definition

Given a measure space a Banach space the Bochner space is defined to be the Kolmogorov quotient (by equality almost everywhere) of the space of all Bochner measurable

is a space of equivalence classes-measure zero As is also usual in the study of such spaces, it is usual to abuse notation rather than an equivalence class (which would be more technically correct).

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 (parametrized by time) of functions of space, possibly in some Bochner space.

Application to PDE theory

is an interval will be one-dimensional Lebesgue measure denotes the Sobolev Hilbert space of once-weakly differentiable that vanish at the boundary of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with compact support denotes the dual space

(The "partial derivative above is actually a total derivative, since the use of Bochner spaces removes the space-dependence.)

See also

References