\n 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 +\n <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
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, […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_joinchat":[]},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts\/227772"}],"collection":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/comments?post=227772"}],"version-history":[{"count":0,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/posts\/227772\/revisions"}],"wp:attachment":[{"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/media?parent=227772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/categories?post=227772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qualityassignments.net\/wp-json\/wp\/v2\/tags?post=227772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}