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].