By Robert E. Megginson

ISBN-10: 0387984313

ISBN-13: 9780387984315

Getting ready scholars for extra learn of either the classical works and present learn, this can be an available textual content for college students who've had a path in actual and complicated research and comprehend the fundamental homes of L p areas. it truly is sprinkled liberally with examples, old notes, citations, and unique assets, and over 450 workouts supply perform within the use of the implications built within the textual content via supplementary examples and counterexamples.

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**Extra resources for An Introduction to Banach Space Theory **

**Example text**

Thus, it is enough to prove that Do =I 0, for then the neighborhood ~Do+ 4{ -D 0 ) of the origin must be included in D. Suppose that D has empty interior. For each positive integer n, the set nD is closed and has empty interior, and so X \ nD is an open set that is dense in X. Let B 1 be a closed ball in X\ D with radius no more than 1. Since (X \ 2D) n Bf is a nonempty open set, there is a closed ball B 2 in B 1 \ 2D with radius no more than 1/2. There is a closed ball B3 in B 2 \3D with radius no more than 1/3.

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. Then IIT(o:1X1 + · · · + O:nXn)ll + · · · + lo:niiiTxnll (11Tx111 + · · · + IITxnll) · lo:1x1 + · · · + O:nXnl, :5lo:1IIITx1ll :5 so T is bounded. This proves the claim. Let I be the identity operator on V, viewed as a member of L(X, W). Since every T in L(X, Y) can be written in the form Tw I where Tw is just T viewed as a member of L(W, Y), the theorem will be proved once it is shown that I is bounded.

Suppose that D has empty interior. For each positive integer n, the set nD is closed and has empty interior, and so X \ nD is an open set that is dense in X. Let B 1 be a closed ball in X\ D with radius no more than 1. Since (X \ 2D) n Bf is a nonempty open set, there is a closed ball B 2 in B 1 \ 2D with radius no more than 1/2. There is a closed ball B3 in B 2 \3D with radius no more than 1/3. Continuing in the obvious way yields a sequence (Bn) of closed balls such that these all hold for each positive integer n: BnnnD = 0, the radius of Bn is no more than 1/n, and Bn 2 Bm if n :::; m.

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

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