By Charles Swartz

ISBN-10: 0824786432

ISBN-13: 9780824786434

In line with an introductory, graduate-level direction given via Swartz at New Mexico nation U., this textbook, written for college kids with a average wisdom of element set topology and integration conception, explains the foundations and theories of useful research and their functions, exhibiting the interpla

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**Extra resources for An introduction to functional analysis**

**Sample text**

Let a, b E R, a < b, and k E IN. Let Ck[a, b] be the subspace of C[a, b] which consists of all functions which have at least k continuous derivatives. Define a norm on Ck[a, b] by k j=0 where f(0) = f. Then Ck[a, b] is a B-space under Example 27. 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...

223). Note I I is not a semi-norm. Example 22. Let (S, E, p) be a measure space and 1 5 p < -. , then LL(p) is a B-space. (This is Riesz's Theorem, [Ro], p. ) Example 16 is a special case of this example where S = IN and p is counting measure ([Ro], p. 55). Example 23. , IIfII°, = p - essensup(f)< -. e. are 1111°,, and if functions which are equal identified, then L°°(p) is a B-space ([Ro], p. 125). In Examples 21, 22 and 23, when I = [a, b] we write LP(I) for Lp(m), where m is Lebesgue measure on I.

Example 16 is a special case of this example where S = IN and p is counting measure ([Ro], p. 55). Example 23. , IIfII°, = p - essensup(f)< -. e. are 1111°,, and if functions which are equal identified, then L°°(p) is a B-space ([Ro], p. 125). In Examples 21, 22 and 23, when I = [a, b] we write LP(I) for Lp(m), where m is Lebesgue measure on I. Example 24. Let a, b E (R, a < b, and let b [a, b] be the space of all b Riemann integrable functions defined on [a, b]. IIf II If I J a semi-norm on ,5E [a, b] which is not complete ([M], p.

### An introduction to functional analysis by Charles Swartz

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