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In mathematics, Bochner spaces are a generalization of the concept of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } of real or complex numbers.
The space L p {\displaystyle L^{p}} consists of all Bochner measurable functions f {\displaystyle f} with values in the Banach space X {\displaystyle X} whose norm ‖ f ‖ X {\displaystyle \|f\|_{X}} lies in the standard L p {\displaystyle L^{p}} space. Thus, if X {\displaystyle X} is the set of complex numbers, it is the standard Lebesgue L p {\displaystyle L^{p}} space.
Almost all standard results on L p {\displaystyle L^{p}} spaces do hold on Bochner spaces too; in particular, the Bochner spaces L p {\displaystyle L^{p}} are Banach spaces for 1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty.}
Bochner spaces are named for the Polish-American mathematician Salomon Bochner.