By Robert E. Megginson

ISBN-10: 0387984313

ISBN-13: 9780387984315

Getting ready scholars for extra examine of either the classical works and present examine, this is often an available textual content for college kids who've had a path in genuine and intricate research and comprehend the fundamental homes of L p areas. it really is sprinkled liberally with examples, historic notes, citations, and unique assets, and over 450 routines offer perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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Notice that ~ is just Euclidean n-space. By c:onvention, the space ~ is Euclidean 0-space. It is also possible to represent ~ as a Lebesgue space by using the fact that there is exactly one scalar-valued function on the empty set, namely, the function represented by 0 x lF; see (65, p. 11, Ex. 3J. ~ can be defined to be Lp(0, {0}, JL), where of course JL(0) = 0. 10 Example: C(K). Let K be a compact Hausdorff space. Then the collection of all scalar-valued continuous functions on K is a vector space under the operations given by the usual addition of functions and multiplication of functions by scalars.

For each member f of this vector space, let ll/lloo = {max{ lf(x)J : x 0 E K} ~f K =f. 0; tf K = 0. It is easy to check that this defines a norm on the vector space. The resulting normed space is denoted by C(K). By analogy with Lp[O, 1], the abbreviation C[O, 1] represents C([O, 1]). 11 Example: rca(K). Let K be a compact Hausdorff space, and let rca(K) be the normed space formed from the collection of all regular finite scalar-valued Borel measures on K by defining the vector space operations in the obvious way and by letting IIJLII be the total variation of JL on K whenever JL is one of these measures.

12 Theorem. Let X andY be normed spaces such that X is finitedimensional. Then every linear operator from X into Y is bounded. PROOF. In this proof, the standard notation 11·11 is used for all norms except the special one about to be defined. Let V be the vector space underlying the normed space X and let n be the dimension of V. Since the theorem is trivial if n = 0, it can be assumed that n ~ 1. •. , Xn be a vector space basis for V. Define a norm I· I on V by the formula . lo:1X1 + · · · + O:nXnl = lo:1l + · · · + Jo:nJ, and Jet W be the normed space (V, I· I).

### An Introduction to Banach Space Theory by Robert E. Megginson

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