By Charles Swartz

In accordance with an introductory, graduate-level path given by way of Swartz at New Mexico country U., this textbook, written for college students with a average wisdom of element set topology and integration idea, explains the foundations and theories of useful research and their purposes, exhibiting the interpla

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35). Finite Products: Let X and Y be quasi-normed spaces. Then X X Y carries natural several I (x, y)12 = quasi-norms. lx, +,Y, and Namely, I (x, Y) 11= I x I + lY , ((x, y)l. = max(lxl, jyj). Each of these quasi-norms induces the product topology on X x Y so any of them can be used on the product. Moreover, if X and Y are semi-NLS, then each of these quasi-norms is a semi-norm and is a norm if and only if X and Y are NLS. A further important example of NLS are given by the inner product and Hilbert spaces.

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. 242). defines a 24 Quasi-normed and Normed Linear Spaces Example 25.

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 (')].