An Introduction to Banach Space Theory by Robert E. Megginson

By Robert E. Megginson

Getting ready scholars for additional examine of either the classical works and present examine, this is often an obtainable textual content for college kids who've had a direction in actual and intricate research and comprehend the fundamental 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 consequences constructed within the textual content via supplementary examples and counterexamples.

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Example text

Let r be a positive numbE)r small enough that the closed ball Br of radius r and center 0 is included in U, and let Mo =sup{ IITxll : x E Br }. If xis a nonzero member of X, then rllxll- 1x E Br, so IIT(rllxll- 1x)ll5 Mo and therefore liTxll 5 r- 1Mollxll. This last inequality also holds if x = 0, so (e) => (f). It is clear that (f) => (g). Finally, suppose that (g) holds. Let M =sup{ liTxll : x E Bx }. Tx- TOll= llTxll 5 t:M, which establishes the continuity ofT at 0. Therefore (g)=> (b), and so (b), (d), (e), (f), and (g) are equivalent.

If a is a scalar, then PROOF. llaTII =sup{ llaTxll : x E Bx} = lal sup{ IITxll : x E Bx} = laiiiTII. If xo E Bx, then II(S + T}(xo)ll $ IISIIIIxoll + IITIIIIxoll $ IISII + IITII, and so liS+ Til= sup{ II(S + T)(x)ll: X E Bx} $ nsn + IITII. Thus, the operator norm is a norm on B(X, Y). Suppose that Y is a Banach space. Let (Tn} be a Cauchy sequence in B(X, Y). If x EX, then IITnX- Tmxll = II(Tn- Tm)(x)ll $ IITn- Tmllllxll 28 1. Basic Concepts whenever n, m E N, from which it follows that the sequence (Tnx) is Cauchy in Y and hence convergent.

Af(s) from S into X are both continuous, that is, that f + g and af are continuous. The verification of the vector space axioms is then easy. _. 0 and that (-f) (s) = - (! (s)) for each f in this vector space and each s in S. • Because of the continuity of the vector space operations, it is possible to develop an interesting theory of series in normed spaces. 5 Definition. Suppose that (xn) is a sequence in a normed space. Then the series generated by (xn) is the sequence ( I::=l x~)==l· For each positive integer m, the mth term I::=l Xn of this sequence of sums is the mth partial sum of the series.

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