Get An Introduction to Banach Space Theory PDF

By Robert E. Megginson

ISBN-10: 0387984313

ISBN-13: 9780387984315

Getting ready scholars for additional 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 genuine and complicated research and comprehend the fundamental homes of L p areas. it's sprinkled liberally with examples, old notes, citations, and unique resources, and over 450 routines supply perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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B) and is therefore a Banach space. 29 Fix a positive real number p. A scalar-valued function f on [0, 1) is said to satisfy a Lipschitz condition of order p if there is a nonnegative real number M such that 1/(s) - f(t)i :::; Mls- tl" whenever s, t E [0, 1). The collection of all such functions is denoted by Lip p. (a) Show that Lip p is a subspace of the vector space of continuous scalar-valued functions on [0, 1). Show that if p > 1, then Lipp consists only of the constant scalar-valued functions on [0, 1).

14 Theorem. Every closed, convex, absorbing subset of a Banach space includes a neighborhood of the origin. PROOF. Let C be a closed, convex, absorbing subset of a Banach space, and let D = Cn( -C). It is enough to show that D includes a neighborhood of the origin. If A is a nonempty subset of D, then 1 1 1 1 1 1 0 E -A+ -(-A) c -D +-(-D)= -D + -D = D 2 2 -2 2 2 2 ' where the last equality comes from the convexity of D. 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.

18 Corollary. For each nonnegative integer n, the only norm topology that IF" can have is its Euclidean topology. 9 shows: However, the different norms must all induce the same· topology. The first of the following two corollaries is obtained by observing that each normed space with finite dimension n must be isomorphic to Euclidean n-space, a Banach space. The second then follows from the first, since each complete subset of a metric space is closed in that space. 19 Corollary. Every finite-dimensional normed space is a Banach space.

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