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In line with an introductory, graduate-level direction given through Swartz at New Mexico nation U., this textbook, written for college kids with a average wisdom of element set topology and integration concept, explains the rules and theories of useful research and their functions, displaying the interpla

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**Example text**

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. 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).

Proof: If T is not continuous at 0, 3 a neighborhood V of 0 in F such that U = T-1 V is not a neighborhood of 0 in E. Then nU is not a neighborhood of 0 do a UV so 3 xn e E with xn I < and xn 9 nU. Then {xn) is bounded (Exer. 2). We will show later that the conclusion of Proposition 4 holds for a large class of spaces called bornological spaces (§21). Chapter 5 45 For semi-NLS, we have Proposition 5. Let X, Y be semi-NLS and T : X -+ Y linear. The following are equivalent. (i) T is uniformly continuous.

I P)1/P < 00. Ip is a B-space under the norm j=1 22 II Quasi-normed and Normed Linear Spaces lip (this is a special case of Example 22). Example 17. Let 0 < p < 1. lp consists of all scalar sequences { tj } 00 satisfying I (t ) I = P IIp < °° IIp defines a quasi-norm on l p (the j=1 la + b I p S I a I p+ I b I p, triangle inequality follows f r o m the inequality 0 < p < 1). Under IIp, l p is a complete, Hausdorff quasi-normed space (Exer. 6). Example 18. Let m0 be the subspace of 100 consisting of all sequences with finite range.