We analyze the dual Ramsey theorem for k partitions and ℓ colors (DRTℓk) in context of reverse math, effective analysis, strong reductions. Over RCA0, stated Baire colorings Baire-DRTℓk is equivalent to statement clopen ODRTℓk a purely combinatorial CDRTℓk. When Borel k≥3, resulting principles are essentially relativizations For each α, there computable code Δα0-coloring such that any partition...

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

We conclude the discussion of additivity, Baire number, uniformity and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cicho´n's diagram.

We prove a parametric version of a smooth convex variational principle with constraints using a Baire category approach. We examine in depth the necessity of the assumptions of our variational principle by providing counterexamples.

A Banach-Steinhaus theorem for sets of continuous linear mappings on topological modules which are Baire spaces is proved, and some consequences of it are derived. Mathematics Subject Classification: 46H25, 46S10, 46E10

Ferrando and Lüdkovsky proved that for a non-empty set Ω normed space X, the c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if only X is, respectively, Baire-like. When metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then over field K (of real complex numbers) all functions f:Ω→X such each ε>0 n∈N ω∈Ω:f(ω)n>ε finite ...

We show that an infinite-dimensional complete linear space X has: • a dense hereditarily Baire Hamel basis if |X| ≤ c; • a dense non-meager Hamel basis if |X| = κ = 2 for some cardinal κ. According to Corollary 3.4 of [BDHMP] each infinite-dimensional separable Banach space X has a non-meager Hamel basis. This is a special case of Theorem3.3 of [BDHMP], asserting that an infinite-dimensional Ba...

As usual, we equip an infinite-dimensional separable Hilbert space 2 by such nonzero σ -finite Borel measures which are invariant with respect to everywhere dense vector subspaces and study duality between such measures and Baire category. Section 1 contains constructions of nontrivialσ -finite Borel measures, which are defined in the infinite-dimensional separable Hilbert space 2 and are invar...

We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces. Any set F of real valued functions defined on an arbitrary set X is partially ordered by the pointwise order, that is f ≤ g iff f(x) ≤ g(x) for all x ∈ X. In other words put f < g iff f(x) ≤ g(x) for all x ∈ X and f(x) 6= g(x) for at least one x ∈ X. Our aim will be to investigate the possibl...