By Charles Swartz

In keeping with an introductory, graduate-level path given by way of Swartz at New Mexico kingdom U., this textbook, written for college kids with a average wisdom of aspect set topology and integration thought, explains the foundations and theories of sensible research and their purposes, displaying the interpla

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12). 3 r> 0 such that B c rV. Pick n E IN such that n >_ r. 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.

As in Example 3. A sequence (fk) converges to 0 if and only if the sequence (Dafk) converges uniformly on compact subsets of SZ V multi-index a. Thus, S(S2) is complete ([DeS], p. 130). ) chosen. Example 29. , xn) . j=1 We will show later that all of these norms are, in some sense, equivalent. Example 30. Let S # 0 and let s be a o algebra of subsets of S. ba(s) denotes the space of all bounded, finitely additive set functions v : s -e ER. 5). 7). ca(s) denotes the subspace of ba(s) which consists of all countably additive measures.

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. Proof: s: Every neighborhood of 0 contains (0) so (0) is bounded. Suppose 3 a E L\{0}. Then na e L t/n. But (n)na = a f+ 0 since a f (0). So L is not bounded. Recall that a subset metrically bounded if S of a metric space (X, d) is said to be 3 a e X, r> 0 such that S c S(a, r), where S(a, r) = (x : d(a, x) < r) is the sphere with center at a and radius r.