In this paper we discuss some fundamental results in real and functional analysis including the Riesz representation theorem, the Hahn-Banach theorem, and the Baire category theorem. We also discuss applications of these theorems to other topics in analysis.

Proof One follows the argument in [1, pp. 432 – 433] (cf. also [2, Sec. 9]) to show that for each fixed badly approximable (k − 2)-tuple (α1, . . . , αk−2) there is a set of second Baire category of (αk−1, αk) ∈ T2 such that conditions (i), (ii), and (iii) of Theorem 1.7 hold for α = (α1, . . . , αk). Because the set of badly approximable (k − 2)-tuples is dense in Tk−2, and the set of second B...

We review basics concerning metric spaces from a contemporary viewpoint, and prove the important Baire category theorem, for both complete metric spaces and locally compact Hausdorff [1] spaces.

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers P not Gδ · Fσ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ P of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. T...

We prove theorems of the following form: if A ⊆ R is a “big set”, then there exists a “big set” P ⊆ R and a perfect set Q ⊆ R such that P × Q ⊆ A. We discuss cases where “big set” means: set of positive Lebesgue measure, set of full Lebesgue measure, Baire measurable set of second Baire category and comeagre set. In the first case (set of positive measure) we obtain the theorem due to Eggleston...