By Robert E. Megginson

Getting ready scholars for additional learn of either the classical works and present learn, this is often an obtainable textual content for college students who've had a path in genuine and intricate research and comprehend the fundamental homes of L p areas. it's sprinkled liberally with examples, historic notes, citations, and unique resources, and over 450 workouts offer perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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21. 30 Let K be a compact Hausdorff space and let X be a normed space. 4, the collection of all continuous functions from K into X is a vector space when functions are added and multiplied by scalars in the usual way. :f: 0; 1f K 0 = 0. The resulting normed space is denoted by C(K, X). (a) Show that ll·lloo is in fact a norm on C(K, X). (b) Show that if X is a Banach space, then so is C(K,X). En x~ be a formal series in a Banach space. (a) Prove that if the series is absolutely convergent, then it is unconditionally convergent.

Furthermore, the number IITII is the smallest nonnegative real number M such that IITxll $ Mllxll for each x in X. Define 1: X -+IR by the formula l(x) = IITxll. Since I is continuous, its supremum on the open unit ball of X is the same as its supremum on the closure Bx of that ball, which gives (a). If x is a nonzero element of Bx, then llxll- 1 x E Sx and l(llxll- 1 x) = llxll- 1 l(x) ~ l(x). It follows that the supremum of I on Sx is the same as its supremum on Bx whenever X "::/: {0}, which gives (b).

F(s),g(s)) ~ (! -. -. af(s) from S into X are both continuous, that is, that f + g and af are continuous. The verification of the vector space axioms is then easy. _. 0 and that (-f) (s) = - (! (s)) for each f in this vector space and each s in S. • Because of the continuity of the vector space operations, it is possible to develop an interesting theory of series in normed spaces. 5 Definition. Suppose that (xn) is a sequence in a normed space. Then the series generated by (xn) is the sequence ( I::=l x~)==l· For each positive integer m, the mth term I::=l Xn of this sequence of sums is the mth partial sum of the series.