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for a measure space ( X, M, u) and 0 < p < 1, define L'( X, u) to be the collection ofmeasurable 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) Definep(f, g)=XIf - 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 thatthere is a measurable subset Xo of X for which u( X~Xo) = 0 and for each < > 0, there is anindex N for whichIfn - fml < < on Xo for all n, m > 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 coFor 1 < p < oo, let f belong to Lo( X, .where q is conjugate of p. Define the linearfunctional Te: LP( X. u) -> R bv4Page 410 / 516-Q +
