By Charles Swartz

ISBN-10: 0824786432

ISBN-13: 9780824786434

In keeping with an introductory, graduate-level path given through Swartz at New Mexico country U., this textbook, written for college students with a average wisdom of aspect set topology and integration idea, explains the foundations and theories of useful research and their functions, exhibiting the interpla

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However, there is one annoying property of ,%' convergence; namely, a X convergent sequence needn't be X bounded as the following example shows. Chapter 4 41 Example 9. Let {tk) Eel with tk > 0 `dk. 18) by II (sk)II = I I tksk I (11 11 clearly depends on { tk} ). k=1 Consider the sequence {ek} in m0. :jEPI}II = I Itn,I} j=k+1 J so {ek} is I I II-X convergent to 0. However, no subseries of converge to an element of m0 with respect to II II II II k ek can since convergence in clearly implies coordinatewise convergence.

If q(0) = 0, then I x = 0. 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].

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.

### An introduction to functional analysis by Charles Swartz

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