Download An introduction to functional analysis by Charles Swartz PDF

By Charles Swartz

ISBN-10: 0824786432

ISBN-13: 9780824786434

In line with an introductory, graduate-level path given via Swartz at New Mexico country U., this textbook, written for college students with a reasonable wisdom of aspect set topology and integration concept, explains the rules and theories of practical research and their purposes, displaying the interpla

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Q(y) V q(z). 2k=2q(x). Then (Uk), x + y + z e Uk_l. Thus, We now have established the conditions of Lemma 2 and we define q(x) 5 x by the formula in Lemma 1. We claim that q(x). The n Suppose x = second inequality follows from the definition of xk, k=1 By Lemma 2 n n q( xk) _ q(xk) ? k=1 2 q(x) k=1 This yields the first inequality. It now follows from the above that xn - 0 in X if and only if I x n1 _40' Since the scalar multiplication on X is continuous, it follow from this that is a quasi-norm on X, and induces the original topology of X.

A semi-normed linear space (semi-NLS is an ordered pair (X, 11 where 11 similarly. 11 is a semi-norm on X. Normed linear spaces ( S) are defined If the semi-norm is understood, we often say that X semi-NLS (or NLS). If (X, 11 11) 11) is a is a semi-NLS (NLS), then d(x,y)=IIx-yli defines a translation invariant semi-metric (metric) on X, and we always assume that a semi-NLS is equipped with this metric topology. A NLS Chapter 2 17 which is complete in this metric topology is called a Banach space or a B-space in honor of the Polish mathematician, Stefan Banach.

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. Example 16. Let 1 5 p < Co, lp consists of all scalar sequences (tj } CO satisfying II {t } IIp = ( I t.

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