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for a measure space ( X, M, u) and 0 lt; p lt; 1, define L'( X, u) to be the collection of
measurable functions on X for which | f| is integrable. Show that LP(X, u) is a linear space.
For f E LP( X, u), define II fIlp = fx IfI P du.
(i) Show that, in general, Il . Ilp is not a norm since Minkowski’s Inequality may fail.
(ii) Define
p(f, g)=
X
If – gl du for all f, g E LP( X, u).
Show that p is a metric with respect to which LP( X, u) is complete.
5. Let ( X, M, u) be a measure space and { fn} a Cauchy sequence in Lo( X, u). Show that
there is a measurable subset Xo of X for which u( X~Xo) = 0 and for each lt; gt; 0, there is an
index N for which
Ifn – fml lt; lt; on Xo for all n, m gt; N.
Use this to show that L ( X, u ) is complete.
19.2 THE RIESZ REPRESENTATION THEOREM FOR THE DUAL OF L'(X, u), 1 = p s co
For 1 lt; p lt; oo, let f belong to Lo( X, .
where q is conjugate of p. Define the linear
functional Te: LP( X. u) -gt; R bv4
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