By Robert E. Megginson

Getting ready scholars for extra examine of either the classical works and present learn, this is often an obtainable textual content for college students who've had a direction in genuine and intricate research and comprehend the elemental houses of L p areas. it truly is sprinkled liberally with examples, ancient notes, citations, and unique assets, and over 450 workouts supply perform within the use of the implications constructed within the textual content via supplementary examples and counterexamples.

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

**Example text**

In any case, the context should always prevent confusion. 7 Proposition. Let X and Y be normed spaces. (a) If En Xn converges in X, then Xn--+ 0. (b) ,If Ln Xn and Ln Yn both converge in X, then so does Ln(xn +Yn), and Ln(Xn + Yn) = Ln Xn +En Yn· (c) If Ln Xn converges in X and a is a scalar, then Ln axn converges, and En axn = a En Xn· (d) If Ln Xn converges in X and Tis a continuous linear operator from X into Y, then En Txn converges in Y, and En Txn = T (En Xn). (e) If Ln Xn is a finite or infinite sum in X, then liEn Xnll ~ Enllxnll· PROOF.

Define T: X -+ Y by the formula Tx = limn Tnx. Because the vector space operations of Y are continuous, the map T is linear. To see that T is bounded, first notice that the boundedness of the Cauchy sequence (Tn) gives a nonnegative M such that IITnll ~ M for each n, so that IITnxll ~ M for each x in Bx and each n. Letting n tend to oo shows that IITxll ~ M for each x in Bx, so T is bounded. To see that IITn - Til -+ 0 as n -+ oo, let e be a positive number and let Ne be a positive integer such that IITn- Tmll ~ e whenever n,m 2::: Ne.

1. The purpose of this section is to derive a few basic properties of linear operators between normed spaces, with special emphasis on the continuous ones. Recall that a subset of a metric space is bounded if it is either empty or included in some ball. ) Each compact subset K of a metric space is bounded, since the metric space is either empty or the union of an increasing sequence of open balls which thus form an open covering for K, fo;rcing K to lie inside one of the balls. Also, every Cauchy sequence in a metric space is bounded, since the sequence from some t~rm onward lies inside a ball of radius 1, and increasing the radius of that ball, if necessary, causes it to engulf the rest of the terms.