Bochner space

In mathematics, Bochner spaces are a generalization of the concept of L^p 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 L^p(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 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 ${\displaystyle 1\leq p\leq \infty }$.

Bochner spaces are named for the Polish-American mathematician Salomon Bochner.

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 ${\displaystyle g(t,x)}$ is a scalar function of time and space, one can write ${\displaystyle (f(t))(x):=g(t,x)}$ to make f a family f(t) (parametrized by time) of functions of space, possibly in some Bochner space.

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