Bochner space

In mathematics, Bochner spaces are a generalization of the concept of ${\displaystyle L^{p}}$ spaces to functions whose values lie in a Banach space which is not necessarily the space ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ of real or complex numbers.

The space ${\displaystyle L^{p}(X)}$ consists of (equivalence classes of) all Bochner measurable functions ${\displaystyle f}$ with values in the Banach space ${\displaystyle X}$ whose norm ${\displaystyle \|f\|_{X}}$ lies in the standard ${\displaystyle L^{p}}$ space. Thus, if ${\displaystyle X}$ is the set of complex numbers, it is the standard Lebesgue ${\displaystyle L^{p}}$ space.

Almost all standard results on ${\displaystyle L^{p}}$ spaces do hold on Bochner spaces too; in particular, the Bochner spaces ${\displaystyle L^{p}(X)}$ are Banach spaces for ${\displaystyle 1\leq p\leq \infty .}$

Bochner spaces are named for the mathematician Salomon Bochner.