REPRESENTATION THEOREMS ON BANACH FUNCTION SPACES 3
differing only on (a-null sets will be identified so that the
elements of M are in reality equivalence classes of functions.)
DEFINITION 1. A mapping p on M to the extended reals
is called a function norm if p satisfies the following conditions
for all f and g in M :
i) p(f) ; 0 and p(f) » 0 if and only if f = 0
ii) p(otf) = ocp(f) for a ; 0
iii) p(f + g) ^ p(f) + p(g)
iv) f £ g in M implies p(f) p(g) .
The definition of p is extended to M , the collection of
all complex valued functions on Q , by defining p(f) = p(|f|)
for f M . In order to avoid pathological cases, only non-
trivial p will be considered, i.e. we require that there exists
f e M such that 0 p(f) ° ° .
DEFINITION 2. i) A function norm p has the weak Fatou
property (WFP) if f tf and sup P(fn) ° ° imply p(f) «
where f M+ and f M+, n = 1,2,... .
n
ii) A function norm p has the strong Fatou property
(SFP) if f tf implies p(fn)tp(f) where f
M+
and fn
M+,
n « 1,2,... . Note that SFP implies WFP, but the converse need
not be true.
DEFINITION 3. Let Lp(Q,Z,[i) = {f M| p(f) « } .
It is clear that L is a normed linear space and, in fact,
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