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FINITE DIMENSIONAL SUBSETS OF INFINITE DIMENSIONAL SPACES by * David W. Henderson This paper gives a summary of some of the results related to the Question: Which spaces of dimension more than n have subsets of dimension n ? (Throughout this paper, "space" will mean "separable metric space.") We first state a few appropriate definitions. Let n always de­ note a non-negative integer. The empty set has dimension -1 (written dim 0 = -i) . A space X has dimension < n (dim X < n ) , if X has a neigh­ borhood basis composed of open sets each with boundary of

ON HOMEOMORPHISMS OF CERTAIN INFINITE DIMENSIONAL SPACES Raymond Y. T. Wong 1. Isotopy Theorem. In this paper we concern only separable metric space X, especially when X is the Hilbert cube I°°, the countably in­ finite product of open unit-intervals 100 or the Hilbert space £2- Let X°° denote the countably infinite product of X by itself. A homeomorphism on X means a homeomorphism of X onto itself. Let G(X) denote the group of homeomorphisms on X. For f,g e G(X), we write f ~ g if f is isotopic to g, by which we mean there exists a map­ ping F of X x I

SOME QUESTIONS IN THE DIMENSION THEORY OF INFINITE DIMENSIONAL SPACES by DAVID W. HENDERSON In this paper, ‘space’ means ‘compact metric space’. The (large in­ ductive) dimension, Ind(X), of a space, X, is defined by transfinite induc­ tion as follows: (a) Ind(X) = -1 , if X is the empty set. (b) For each ordinal a, Ind(X) < a, if each closed subset of X has arbitrarily small neighborhoods whose boundaries have dimension < a, (c) Ind (X) = a, if Ind (X) < a and if, for each /3 < a, it is not true that Ind (X) < /3. (d) If Ind (X) either fails to exist or is

Appendix III Holomorphic Functions on Infinite-Dimensional Spaces This appendix contains some facts on holomorphic functions between domains in infinite-dimensional spaces and complex manifolds. For a detailed discussion of this topic, we refer to Herve's book [He89] (see also the appendix in [Lan75]). Preliminaries on holomorphic functions Definition A.III.l. Let X and V be sequentially complete locally convex (s.c.l.c.) spaces and U C X open. (a) A function /: U —> V is called Gateaux holomorphic ((G)-holomorphic) if for each finite-dimensional affine subspace

Adv. Calc. Var. 5 (2012), 59–76 DOI 10.1515/ACV.2011.010 © de Gruyter 2012 Perimeter of sublevel sets in infinite dimensional spaces Vicent Caselles, Alessandra Lunardi, Michele Miranda jr and Matteo Novaga Communicated by G. Mingione Abstract. We compare the perimeter measure with the Airault–Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains. Keywords. Perimeter, abstract Wiener spaces, convex sets

Ergodicity of Markov semigroups on infinite dimensional spaces Boguslaw Zegarlinski Abstract. The recent progress in understanding the ergodicity problem for Markov semigroups on infinite dimensional spaces is presented. 1991 Mathematics Subject Classification: 82B20, 60K35 1. The ergodicity problem Let μ be a probability measure on (Ω,Σ), a Polish infinite dimensional space with its Borel σ-algebra. Let Pt — etC, t > 0 be a Markov semigroup on the Β an ach space (£(Ω), || · | |u} of continuous functions with the supremum norm, such that ßfPtQ = ßgPtf

PROGRAMMING IN INFINITE DIMENSIONAL SPACE AND CONTROL THEORY H. Halkin Introduction. The a i m of this l ec ture is to p r e s e n t some n e c e s s a r y conditions for a g e n e r a l type of matheraat ica l p r o b l e m s in infinite dimen­ sional space. Our main motivation in establishing those n e c e s s a r y con­ ditions was to obtain a resu l t which would include, as p a r t i c u l a r c a s e s , the s tandard n e c e s s a r y conditions of calculus of var ia t ions and optimal control . The resu l t s given h e r e w e r e f i rs t p r e s e

11 A little more about convex cones in infinite-dimensional spaces Throughout this section we will assume that X is a Banach space and K is a convex closed cone in it, such that K ̸= X. In this situation we will be interested in the case when the space X is infinite-dimensional since otherwise the results given below im- mediately follow from the finite-dimensional separability of convex sets.¹ The following question is of great interest for applications: whether there exists a linear continuous functional l ̸= 0 on X which is nonnegative on the cone K, i