The weak Banach-Saks Property of the Space ( L p μ ) m
Abstract
In this paper we show the weak Banach-Saks property of the Banach vector space
(
L
p
μ
)
m
generated by
m
L
p
μ
-spaces for
1≤p<+∞,
where
m
is any given natural number. When
m=1,
this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of a vector space
R
m
and vector-valued functions in a weakly compact subset of the space
(
L
p
μ
)
m
for
1≤p<+∞
and inequalities when these vector-valued functions are in a weakly* compact subset of the product space
(
L
∞
μ
)
m
generated by
m
L
∞
μ
-spaces.