Download e-book for kindle: An Introduction to Banach Space Theory by Robert E. Megginson

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 can be an obtainable textual content for college students who've had a direction in actual and intricate research and comprehend the elemental homes of L p areas. it's sprinkled liberally with examples, historic notes, citations, and unique assets, and over 450 workouts offer perform within the use of the implications constructed within the textual content via supplementary examples and counterexamples.

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Sample text

It is easy to check that ll·llao is a norm. The resulting normed space is called lao (pronounced "little ell infinity"). A convention used for lao and other spaces whose elements are sequences is that the phrase "a sequence in the space" always means a sequence of elements of the space, never a single element. Suppose that JL is the counting measure on the collection E of all subsets of N, that is, the measure given by letting JL(A) be the number of elements in A. Then the measurable functions from N into lF are just the sequences of scalars, and Lao(N, E, JL) is lao.

Suppose that S, T E B(X, Y). It is clear that IITII ~ 0. If T "::/: 0, then there is an xo in X, necessarily nonzero, such that Txo "::/: 0, and so T(llxoll- 1x 0 ) "::/: 0. It follows that T = 0 if and only if Tx = 0 for each x in Bx, that is, if and only if II Til = 0. 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.

An important technique was used in the proof of the preceding theorem. To prove that a linear operator T from X into Y must be continuous, it was shown that there is a normed space W such that all the linear operators from W into Y have this property, and that T can be factored as the product of a continuous linear operator from X into W and a linear operator from W into Y. Variations of this technique with other properties besides continuity have many applications in Banach space theory. 13 Definitions.

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An Introduction to Banach Space Theory by Robert E. Megginson


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