Orlicz Spaces and Modular Spaces
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~[ODULAR
SPACE S
§ 1o~lodular spaces 1,1~Definitiono Let X be a real or complex vector space. A functional
~ :X-,~O,~] is called a pseudomodular,
if there hoTds for arbi-
trary x,y e X : 1°
~ LO) = O,
2°
~(-x)= ~Cx~ in case of X real ~(eitx)= ~(x) for every real t in case of X complex,
3°
~(~x+~y).~
~(x~+ ~(y) for
~,~O,
~+~=
I.
If in place of 3 ° there holds 3° • with an s6(0,I~ , then the pseudomodular
~ is called s-convexo l-con-
vex pseudomodulars are called con~exo If besides I ° there holds also 10" then
~ ( ~ x ) = O for all
A>O
implies x = O,
~ is called a semimodular. If moreover, I0 "
then
~(X)= 0 implies x = O,
~ is called a modular. Io2.Exampleso I® If X is an s-normed space with an s-homogeneous
norm
Jl IIs, then
9(x)=J~ll s is an s-convex modular in X. Similarly, if
Jl II s is an s - h o m o ~ n e o u s
pseudonorm in X, then
~(x)=}Ixll s is an s-con-
vex pseudomodular in Xo However, in general a modular is not a norm~ among others, it may assume the value +co. IIo Let X = L p over the interval p-convex modular on X for O < p
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