Embeddings and Extensions in Analysis by J.H. Wells, L.R. Williams

By J.H. Wells, L.R. Williams

The item of this e-book is a presentation of the foremost effects when it comes to geometrically encouraged difficulties in research. One is that of picking out which metric areas should be isometrically embedded in a Hilbert house or, extra in most cases, P in an L area; the opposite asks for stipulations on a couple of metric areas as a way to make sure that each contraction or each Lipschitz-Holder map from a subset of X into Y is extendable to a map of an identical sort from X into Y. The preliminary paintings on isometric embedding was once all started by means of okay. Menger [1928] together with his metric investigations of Euclidean geometries and persevered, in its analytical formula, through I. J. Schoenberg [1935] in a sequence of papers of classical attractiveness. the matter of extending Lipschitz-Holder and contraction maps used to be first taken care of through E. J. McShane and M. D. Kirszbraun [1934]. Following a interval of relative state of no activity, consciousness used to be back attracted to those difficulties through G. Minty's paintings on non-linear monotone operators in Hilbert area [1962]; by way of S. Schonbeck's basic paintings in characterizing these pairs (X,Y) of Banach areas for which extension of contractions is usually attainable [1966]; and by way of the generalization of a lot of Schoenberg's embedding theorems to the P atmosphere of L areas via Bretagnolle, Dachuna Castelle and Krivine [1966].

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Extra resources for Embeddings and Extensions in Analysis

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Our opening remarks establish the conclusion in case 2 < p ~ 00. §8. 12). 9). To show that every function in N(U) (0 < p ~ 2) has the indicated form we need the following lemma: Consider the sequence qJn(r) = 1 00 1_e-ru u o (0 ~ r) dPn(u) where Pn is a positive measure on IR+ and (n= 1,2, ... ). If cpir) ---+ cp(r) for each re IR+ and cp is continuous at zero, then cp(r) = 1_e- ru dP(u) o u frO (0 ~ r) where P is a positive measure on IR+ such that roo J1 dP(u) < 00. u The sequenceYn of positive measures on [0,00 ] defined by dYn(u) = 1- e- U dPn(u) u is uniformly bounded since J~ dYn(u) = CPn(1) ---+ cp(1) as n ---+ 00.

13) remain unchanged. Thus we can apply Stone's theorem to conclude the existence of a resolution of the identity E on the Borel subsets of IRN and a densely defined self-adjoint operator A on HI. such that A= Ie N x dE(x) and U(s) = exp(isA) = JiliN e'(S,JC)dE(x) (se IR N). Choose 0 < 0 < 1 and let 1 1 ,1 2 ,1 3,... ,lm be a partItion of SN-l into disjoint Borel sets, each with nonempty interior if N > 1, and such that l(u,v)1 ~ 0 for all u, vel j for j=I,2, ... ,m. Then partition IRN into the sets 100 = to} and IJk = {xeIRN: x/llxlleIj and 2- k - 1 < Ilxll ~ 2- le } U=1, ...

7) §7. 8) ~O). 9) . 2. oiru) da(u) (0 :s; r < (0). 11) Proof. on shares the same properties. 4(d). For the converse, suppose that FeRPD(lRn). 12) (xe IRn) for some finite positive measure v on IR n. Fix r ~ o. Since f(re) is constant as ~ ranges over Sn-1, we may write J F(r) = f«r,O, ... _ 1 ei(PU)dun_1(~)} dv(u) (r ~ 0). The reduction in the last step from an integral over IR n to an integral from 0 to 00 is possible because the integrand is radial. The definition of a on Borel subsets G of [0,(0) is a(G) =v({xe 1R":llxlleG}).

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