Bochner space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Bochner spaces are a generalization of the concept of <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces</a> to functions whose values lie in a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> which is not necessarily the space 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 or 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 of real or complex numbers.
The space 
 
 
 
 
 L
 
 p
 
 
 (
 X
 )
 
 
 {\displaystyle L^{p}(X)}
 
 consists of (equivalence classes of) all <a href="/facts/Bochner_measurable/HxXFeAgt">Bochner measurable</a> functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with values in the Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 whose <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> 
 
 
 
 ‖
 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
 
 
 (
 X
 )
 
 
 {\displaystyle L^{p}(X)}
 
 are Banach spaces for 
 
 
 
 1
 ≤
 p
 ≤
 ∞
 .
 
 
 {\displaystyle 1\leq p\leq \infty .}

Bochner spaces are named for the <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a> <a href="/facts/Salomon_Bochner/n35gmr7h">Salomon Bochner</a>.

Bochner space open-in-new

Bochner space