By Charles Swartz

In keeping with an introductory, graduate-level path given via Swartz at New Mexico nation U., this textbook, written for college students with a average wisdom of element set topology and integration concept, explains the foundations and theories of sensible research and their functions, displaying the interpla

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Theorem 3 (Kakutani). The topology of a first countable TVS X is given by a quasi-norm. Proof: Put U0 = X and choose a neighborhood base at 0 Un+i + Un+1 + Un+1 c U. balanced sets, (Un)n=1, satisfying of For X E (0), set q(x) = 0, and for x e (0) set q(x) = 2 k, where k = k(x) is the largest integer such that x E Uk [thus, if k = k(x), x E Uk\Uk+l]. Note q(x) = q(-x) since the Uk s are balanced. We claim that xn - 0 in X if and only if q(xn) -40. Suppose xn -40 and let m be a positive integer. Then xn E Um eventually and for such xn either xn E (0) so q(xn) = 0 or k(xn) ?

R convergent to an element of X. 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.

Proposition 10. if-space, then X is a X-space. Proof. Let xk -+ 0 in X. 5). is bounded (Exer. 5) and, therefore, X bounded. Thus, {t k(tkxk) } = { xk } is X convergent. Bounded Sets in a TVS 42 Exercise 1. Show that a finite set is bounded. Exercise 2. Show a compact subset of a TVS is bounded. Exercise 3. Show that a subset of a bounded set is bounded. Exercise 4. Show that a finite union or (vector) sum of bounded sets is bounded. Exercise 5. Show that a convergent sequence is bounded. Exercise 6.