Read e-book online Geometric Functional Analysis and its Applications PDF

By Richard B. Holmes (auth.)

ISBN-10: 1468493698

ISBN-13: 9781468493696

ISBN-10: 146849371X

ISBN-13: 9781468493719

This ebook has developed from my adventure over the last decade in instructing and doing examine in sensible research and likely of its appli­ cations. those functions are to optimization thought ordinarily and to top approximation idea particularly. The geometric nature of the themes has vastly motivated the method of sensible research awarded herein, specifically its foundation at the unifying thought of convexity. lots of the significant theorems both predicament or rely on houses of convex units; the others more often than not pertain to conjugate areas or compactness homes, either one of which subject matters are vital for the right kind environment and backbone of optimization difficulties. consequently, and unlike such a lot different remedies of practical research, there is not any dialogue of spectral thought, and purely the main uncomplicated and basic houses of linear operators are validated. many of the theoretical highlights of the publication are the Banach area theorems linked to the names of Dixmier, Krein, James, Smulian, Bishop-Phelps, Brondsted-Rockafellar, and Bessaga-Pelczynski. sooner than those (and others) we determine to 2 most vital ideas of geometric sensible research: the prolonged Krein-Milman theorem and the Hahn­ Banach precept, the latter showing in ten diverse yet an identical formulation­ tions (some of that are optimality standards for convex programs). additionally, a great deal of realization is paid to homes and characterizations of conjugate areas, specially reflexive spaces.

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Thus if X is (locally) contractible then X is (locally) path connected and (locally) simply connected. It is clear that a linear topological space is both contractible and locally contractible, since the map (a, x) f-+ ax is continuous by definition. Finally, some linear topological spaces of considerable importance in our subject (conjugate spaces with the "weak-star" topology; defined later) turn out to be expressible as a countable union of compact subsets. Such spaces then have the Lindelof covering property (every covering by open subsets admits a countable subcovering).

We have already noted that a linear topological space X must be homogeneous, that is, given Xb X 2 E X, there exists a homeomorphism h:X ---+ X such that h(xd = X2. ) This follows from 9A using that X contains a local base of closed sets. Let us recall that a topological space X is contractible if the identity map on X is nullhomotopic. This means intuitively that X can be continuously shrunk to a point, and precisely that there is a continuous function (a homotopy) h:[O, 1] x X ---+ X such that h(O, x) = x (x E X) and h(1, x) is constant.

Linear Topological Spaces In this section we give the definition and fundamental properties of "linear topologies". This notion is too general for our purposes and it will shortly be specialized by the introduction of a geometrical constraint on the basic neighborhoods. A. Let X be a linear space over the (real or complex) scalar field F. We recall that a topology on X is a family 5 of subsets of X, closed under the formation of finite intersections and arbitrary unions, and containing in particular the empty set 0 and the whole space X.

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