REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES 5
The constant Y = 1 if and only if p has SFP.
DEFINITION 7. The dual norm p' of a function norm p is
defined by p'(f) = sup {JQ | fg| dp. I p(g) s 1} . Then the dual
space is defined as L / = {f M|p'(f) °° } .
REMARKS 8. The following are some useful facts about dual
spaces:
i) p' is a function norm with SFP and L / is a
Banach function space.
ii) There are no unfriendly sets relative to p'
iii) There exists a sequence {Q } that is jointly p
and p' admissible.
iv) Higher duals are similarly defined, e.g.
p"(f) = sup {J |fg|d|i I p'(g) 1} .
Q '
v) If y is the constant of theorem 6, then
YP(f) ^ p"(f) * p(f) for all f L ; i.e. L and L » have
the same functions and equivalent norms, and they are identical
if and only if p has SFP.
vi) The regular and inverse Holder inequalities hold
in the following forms: If f L and g L / , then fg L.
and |jfg|U ^ P(f)p7(g) Conversely, if fg 6 L for all g L /
then f £ L . In fact, the Inverse Holder inequality holds with
the weakened hypothesis that fg L- , for all g in a closed
norm-determining subspace of L / . This is not stated explicitly
in [19] but is readily seen in the proof that appears there,
which is also the proof found in [18].
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