By Charles Swartz

ISBN-10: 0824786432

ISBN-13: 9780824786434

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

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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.

X = { } E c00 if and only if 3 N (depending on x) such that J t = 0 for j > N. We assume that c00 is equipped with the sup-norm. c00 is not complete with respect to the sup-norm. Let the sequence with a If coordinates. (ti } 1 in the jth coordinate and 0 J be in the other is a scalar sequence which converges to 0 with n o 0, then the sequence sn = Jee is Cauchy but does not converge to j=1 an element of c00. The sequence (tj ) also furnishes an example of a sequence which converges to 0 but which is not X convergent.

We also follow the tradition of writing T(x) = Tx. Proposition 1. Let E, F be TVS and T : E -4 F linear. The following are equivalent. (i) T is continuous. (ii) T is continuous at 0. Proof: Assume (ii) holds and let x e E. Let V be a neighborhood of Tx. Then -Tx + V is a neighborhood of 0 so 3 a neighborhood U of 0 in X such that TU c -Tx + V. Thus, x + U is a neighborhood of x with T(x + U) c V, and (i) holds. 43 Linear Operators and Linear Functionais 44 Definition 2. A linear map between TVS is said to be bounded if it carries bounded sets to bounded sets.

### An introduction to functional analysis by Charles Swartz

by Charles

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