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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