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By Charles Swartz

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

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Then B c nV c nU since V is balanced [rV = --rn(nV) c nV]. Thus, (vi) * (vii). Suppose (vii) holds. Let (xk} c B and tk -40. Let U be a balanced neighborhood of 0. 3 N such that I tk <_ In for k >_ N. Then n xk E U for k >_ N, and since U is balanced, tkxk = (ntk)n xk E U for k >_ N. Hence, tkxk -1 0. Remark 3. Condition (iii) was introduced by von Neumann and is often used for the definition of boundedness ([vN]). Chapter 4 39 Corollary 4. A linear subspace L of a TVS is bounded a L c (0). In particular, no non-trivial subspace of a Hausdorff TVS is bounded.

N n If q(x) = q(-x) and x = n - xk and k=1 k=1 n q(xk) = k=1 so I xk, then -x = q(-xk) k=1 lxI = I-XI. Lemma 2. Let X be a vector space and q a non-negative function of X such that q(0) = 0 and q(x +- y + z) 5 2 max (q(x), q(y), q(z) } . , xn a X, q( xi) S 2 i=1 q(xi). i=1 n Proof: Set a = q(xi), where we may assume a > 0. The proof i=1 is by induction on n. For n = 1, 2, 3, the result is trivial so assume that m n > 3. Let m be the largest integer such that q(xi) 5 a/2 [if this i=1 inequality fails to hold for m = 1, set m = 0 and ignore this inequality].

In Examples 21, 22 and 23, when I = [a, b] we write LP(I) for Lp(m), where m is Lebesgue measure on I. Example 24. Let a, b E (R, a < b, and let b [a, b] be the space of all b Riemann integrable functions defined on [a, b]. IIf II If I J a semi-norm on ,5E [a, b] which is not complete ([M], p. 242). defines a 24 Quasi-normed and Normed Linear Spaces Example 25. Let D c C be an open, connected set, and let J(D) be the (Kn) space of all analytic functions f : D -' C. Let be an increasing sequence of compact subsets of D each of which has non-void interior, 00 Kn = D and each compact subset of D is contained in some Kn.

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