An introduction to functional analysis by Charles Swartz

By Charles Swartz

According to an introductory, graduate-level direction given by way of Swartz at New Mexico kingdom U., this textbook, written for college kids with a reasonable wisdom of element set topology and integration thought, explains the rules and theories of practical research and their purposes, exhibiting the interpla

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We give isometric descriptions of the duals of some of the classical function spaces. It is assumed that the reader is familiar with some of the classic dual spaces from real analysis; references to such duals are given. Example 11. c' and 11 are linearly isometric. Let f E c6. Set yk = for k E IN and y = tyk). We claim Linear Operators and Linear Functionals 48 N that y E ll. , s IIfII k=1 Co so y ell and IIfII I <_ IIfII. x = (xk) E CO, x = For xkek so k=1 C _ xkyk and xk _ k=1 k=1 Co II < IIxII- IYkI = IIxII-11Y111 k=1 which implies that IIfII s 11y111.

U(E n E) for E e I. It is easily checked from the finite j=1 Linear Operators and Linear Functionals 50 additivity of µ that the integral is independent of the representation of tp. We also have (2) I is 0 dµ I s II gpII. I µ I (S), where IµI is the variation of u Every f E B(S, E) is the uniform limit of a sequence of simple functions, (901 so we can define the integral of f with respect to µ to be is f dµ = lim JSqkdµ [the limit exists since by (2), Ifsqkdµ- IS 4jdJI S III - (PjLIAI(S) and, moreover, the value of the limit is independent of the particular sequence (')].

Let K c " be compact. Let 11 IL ,k ([DeS], p. 130). 9)K be all scalar valued functions f : IR' -4F which have continuous partial derivatives of all orders with support contained in K. , an) with n aj a non-negative integer, let I a ' j=1 a , and write Chapter 2 25 Daf = al al f axon... axal n for f e 9JK. 0 K set IIfIlk = for k = 0, 1, .... 0K. 0K is complete under this quasi-norm ([DeS], p. 130). It is not clear that OK contains anything other than the zero function. 0 K. , xn) = gp(x1) ... 0 K) where K = [-1, 1] x ...

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