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Journal: Annales Mathematicae Silesianae

Existence and uniqueness of continuous solutions of nonlinear functional equations are generic properties

Witold Jarczyk

Language: EN | Published: 30-01-1990 | Abstract | pp. 32-40

Fundamental properties of equations of the form (1) are discussed from the Baire category point of view.  After showing that they are generic in a suitable function space the density of the set of equations (1) having no solutions is studied. Results of the paper are “product versions” of these proved in [3].

1990-01-30



Journal: Annales Mathematicae Silesianae

A really simple elementary proof of the uniform boundedness principle in F-spaces

José L. Ansorena

Language: EN | Published: 18-06-2024 | Abstract | pp. 42-45

We give a proof of the uniform boundedness principle for linear continuous maps from F-spaces into topological vector spaces which is elementary and also quite simple.

2024-06-18



Journal: Annales Mathematicae Silesianae

Duality principle of W. Sierpiński in the abstract Baire-cathegory theory

Piotr W. Cholewa

Language: EN | Published: 30-09-1986 | Abstract | pp. 30-36

Let C be an M-family of subsets of X and C₁ - the family of its “first category” sets. It is proven that one and only one of the following conditions is satisfied: (*) each C₁-set is at most countable; (**) X is the union of C₁ set and a set having property (L), which are disjoint; (***) each C-residual set contains an uncountable C₁-set.
Moreover, if C⊂2^X and D⊂2^Y are two M-families, the “duality principle” holds (i.e. there exists a bijection f: X→Y transforming C₁-sets onto D₁-sets) iff C and D satisfy the same of the conditions above.
Also, some considerations are added, concerning the coincidence between the properties of the family C₁ and a σ-ideal.

1986-09-30