# An introduction to functional analysis by Charles Swartz PDF

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 kids with a reasonable wisdom of element set topology and integration idea, explains the foundations and theories of sensible research and their purposes, exhibiting the interpla

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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. Hence, IIfII =11y1I1 Thus, the map U : f --4 y from c' to obviously linear. 11 is an isometry and is The map U is also onto since if y = {yk} E ll, 00 ykxk defines a continuous linear functional with U(fy) = Y.

Exercise 15. If the sequence (xk) is Mackey convergent to 0, show mat it converges to 0. Exercise 16. Show that COO' c, c0, e p (1 <_ p < co) are separable. Show 100 is not separable. Exercise 17. Show B(S, E) of Example 31 is complete. 3 Metrizable TVS If X is a TVS whose topology is semi-metrizable, then the topology is certainly first countable. This simple necessary condition turns out to be sufficient for the topology of a TVS to be semi-metrizable even by a quasi-norm. To establish this result, we require two lemmas.

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 (')]. The inequality (2) still holds if (p belongs to B(S, E). If µ E ba(s), then by (2) µ induces a continuous linear functional fu on B(S, E) with IIfpII <_ I p.