By Charles Swartz

ISBN-10: 0824786432

ISBN-13: 9780824786434

In accordance with an introductory, graduate-level direction given by way of Swartz at New Mexico kingdom U., this textbook, written for college students with a average wisdom of aspect set topology and integration concept, explains the foundations and theories of practical research and their functions, displaying the interpla

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In this case, we write k=1 n 00 If (X, I I) xk is said to be is a quasi-normed space, the series k=1 00 absolutely convergent if the series 1x1 converges. We have the k=1 following criterion for completeness in quasi-normed spaces. Theorem 9. A quasi-normed space (X, I I) is complete if and only if every absolutely convergent series in X is convergent. n Proof: Suppose xk is absolutely convergent and set sn = xk. k=1 n If n > m, I sn - sm I <_ I xk I k=m+1 convergent. so { sn } is Cauchy and, therefore, Chapter 2 19 Conversely, suppose that (xk) is Cauchy in X.

It is actually the case that IIfpII = IJI (S) = IIµII Let c > 0. , n) n of S such that I µ(Ej) I +e. I µ I (S) < Define tp : S -i (R by j=1 n (p = I sign µ(E )CE . Then II T11 = 1 and j=l n

8). Example 12. c is the subspace of l °° consisting of all the convergent sequences. We assume that c is equipped with the sup-norm. We show that c is complete by showing that it is a closed subset of the complete space l Chapter 2 21 Corollary 13. c is a B-space. Proof: Let xn = { t } =1 E c converge to a point x = (t() e Q J >_ I t Let lim t = tn. Since (Ixn - xII. j I `dj, lim t = ti uniformly in j. , and xE c. Thus, urn urn t n J j n J n j J J Example 14. co is the subspace of c consisting of the sequences which converge to 0.

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