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appendixa

# appendixa - A Metrics Norms Inner Products and Topology...

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A Metrics, Norms, Inner Products, and Topology These appendices collect the background material needed for the main part of the volume. In keeping with the philosophy of this text, we formulate these as “mini-courses” with the goal of providing substantial, though not exhaustive, introductions to and reviews of their respective subjects. Topics that are not typically part of standard beginning mathematics graduate courses are given more detailed attention, while other results are either formulated as exercises (often with hints) or stated without proof. Sources for additional information on the material of this and most of the other appendices include Folland’s real analysis text [Fol99], Conway’s functional analysis text [Con90], and the the operator theory/Hilbert space text [GG01] by Gohberg and Goldberg. A.1 Notational Conventions We first review some of the notational conventions that are used throughout this volume. Unless otherwise specified, all vector spaces are taken over the complex field C . In particular, functions whose domain is R d (or a subset of R d ) are generally allowed to take values in the complex plane C . Integrals with unspecified limits are taken over either the real line or R d , according to context. In particular, if f : R C , then we take integraldisplay f ( x ) dx = integraldisplay −∞ f ( x ) dx. The extended real line is R ∪{−∞ , ∞} = [ −∞ , ] . We use the conventions that 1 / 0 = , 1 / = 0 , and 0 ·∞ = 0 . If 1 p ≤∞ is given, then its dual index or dual exponent is the extended real number p that satisfies 1 p + 1 p = 1 .

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224 A Metrics, Norms, Inner Products, and Topology Explicitly, p = p p 1 . The dual index lies in the range 1 p ≤∞ , and we have 1 = , 2 = 2 , and = 1 . The Kronecker delta is δ ij = braceleftBigg 1 , i = j, 0 , i negationslash = j. A.2 Metrics and Convergence A metric determines a notion of distance between points in a set. Definition A.1 (Metric Space). Let X be a set. A metric on X is a function d: X × X R such that for all f, g, h X we have: (a) d( f, g ) 0 , (b) d( f, g ) = 0 if and only if f = g, (c) d( f, g ) = d( g, f ) , and (d) the Triangle Inequality : d( f, h ) d( f, g ) + d( g, h ) . In this case, X is a called a metric space . The value d( f, g ) is the distance from f to g. If we need to explicitly identify the metric we write “let X be a metric space with metric d” or “let ( X, d) be a metric space.” A metric space need not be a vector space, although this will be true of most of the metric spaces encountered in this volume. Once we have a notion of distance, we have a corresponding notion of convergence. Definition A.2 (Convergent and Cauchy Sequences). Let X be a met- ric space with metric d , and let { f n } n N be a sequence of elements of X. (a) We say that { f n } n N converges to f X if lim n →∞ d( f n , f ) = 0 , i.e., if ε > 0 , N > 0 , n N, d( f n , f ) < ε.
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appendixa - A Metrics Norms Inner Products and Topology...

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