The in…nite combinatorics here give statements in which, from some sequence, an in…nite subsequence will satisfy some condition – for example, belong to some speci…ed set. Our results give such statements generically –that is, for ‘nearly all’points, or as we shall say, for quasi all points –all o¤ a null set in the measure case, or all o¤ a meagre set in the category case. The prototypical res...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/05 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-example: non-existence of norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theore...

Introduction. Some years ago Hausdorff, using Hamel bases, showed that any infinite dimensional real Banach space contained a second category linear subspace that was not complete under any equivalent norm [5]. It is shown below that a slight abstraction of his construction leads to the following: (A) a "multiple offender" example (Theorem 3), one that combines several pathological features whi...

We obtain game–theoretic characterizations for meagerness and rareness of filters on ω. One of the classical methods for obtaining a set of real numbers which does not have the property of Baire, is to interpret appropriate filters on N = {1,2,3, . . . } as subsets of [0, 1]. Filters which result in a set having the property of Baire have nice combinatorial characterizations, due to Talagrand [...

The Baire category theorem implies that the family, F,of dense sets G6 in a fixed metric space, X , is a candidate for generic sets since it is closed under countable intersections; and if X is perfect (has no isolated point), then A E F has uncountable intersections with any open ball in X. There is a long tradition of soft arguments to prove that certain surprising sets are generic. For examp...

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire sp...

Let X be a Polish space, that is, a separable, completely metrizable topological space. We always assume that X is non-empty and perfect, that is, X has no isolated points. An old theorem of Mycielski (see [8]) says that given a sequence of meager sets …Bn Í X k…n†: n < q†, where 1 < k…n† < q, there exists a perfect set P Í X such that P is free for each Bn , that is, P k…n† Ç Bn Í f…x0 ; . . ....

Brown and Simpson (1993) considered two versions of the Baire Category Theorem, which they called BCT-I and BCT-II . They showed that BCT-I is provable in RCA0, the subsystem of second order arithmetic specifying some basic algebraic properties of the numbers, induction for Σ 1 sets and the recursive comprehension axiom. Additionally, they showed that BCT-II is not provable in RCA0. Brown and S...

It is known that a real valued measure (1) on the a-ring of Baire sets of a locally compact Hausdorff space, or (2) on the Borel sets of a complete separable metric space is regular. Recently Dinculeanu and Kluvanek used regularity of non-negative Baire measures to prove that any Baire measure with values in a locally convex Hausdorff topological vector space (TVS) is regular. Subsequently a di...