By Robert E. Megginson

ISBN-10: 0387984313

ISBN-13: 9780387984315

Getting ready scholars for additional examine of either the classical works and present learn, this is often an available textual content for college kids who've had a path in actual and intricate research and comprehend the elemental 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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**Additional info for An Introduction to Banach Space Theory **

**Example text**

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). flxn) in W converges to some member o:1x1 + · · · + O:nXn of W if and only if lim; a:~) = O:m when m= l, ... ,n. 30 1. Basic Concepts Claim: If Z is a normed space, then each member of L(W, Z) is bounded. To see this, suppose that T E L(W, Z) and o: 1 x 1 + · · · + O:nXn E W.

14 Example: Lp('ll') and Hp, 1 ~ p ~ oo. Fix p such that 1 ~ p ~ oo. Let 1l' be the unit circle { z: z E C, lzl = 1} in the complex plane, and let X be the set of all complex-valued functions f on 1l' with the property that if g: [-1r, 1r) ~ C is defined by letting g(t) = f(eit), then g E Lp[-1r, 1r). As would be expected, two functions fi and h in X are considered to be the same if JI(eit) = h(eit) for almost all tin [-1r, 1r). Since Lp[-1r, 1r) is a Banach space, it is clear that X is also a Banach space with the obvious vector space operations and the norm ll·llp given by letting 11/llp = { ( 2~ i: ll9lloo lf(eit)JP dt r/P (where g is as above) if 1 ~ p < OOj if p = oo.

Let p be such that 1 5 p :5 oo and let n be a positive integer. Define a norm on the vector space lFn by letting ll(a:l,···•O:n)llp= { ( I)a:;IP) l/p i=l max{la:1l, ... ,la:nl} if 1 :5 p < oo; if p = oo. The resulting normed space is called £; (pronounced "little ell p n" ). 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.

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