Zygfryd Kominek

Boundedness and continuity of additive and convex functionals

SummaryIt is shown that for any real Baire topological vector spaceX the set classesA(X):={T ⊂ χ: for any open and convex setD ⊃ T, every Jensen-convex functional, defined onD and bounded from above onT, is continuous} andB(X):={T ⊂ χ: every additive functional onX, bounded from above onT, is continuous} are equal. This generalizes a result of Marcin E. Kuczma (1970) who has shown the equalityA(ℝn)=B(ℝn) However, the infinite dimensional case requires completely different methods; therefore, even in the caseX = ℝn we obtain a new (and perhaps simpler) proof than that given by M. E. Kuczma.

On Hyers-Ulam stability of the Pexider equation

Let (S, +) be a commutative semigroup and let X be a sequentially complete linear topological Hausdorff space. In the theory of functional equations the problem of the stability (in a sense) has been considered by many authors. We recall only two results concerning the stability of the Pexider equation. In [1] E. Glowacki and Z. Kominek have proved that under the above assumptions, for arbitrary functions f,g,h : S —• X fulfilling the condition

On stability of the homogeneity condition

Let ƒ be a function defined on a cone S with the values in a sequentially complete locally convex linear topological Hausdorff space Y. If there exist a bounded subset V of Y and an open interval (a, b) ⊂ (1,∞) such that for all x ∈ S and every A ∈ (a, b) the condition λ−1 ƒ(λx) − ƒ(x) ∈ V holds, then there exists a unique positively homogeneous mapping F: S → Y such that the difference F(x) − ƒ(x) is uniformly bounded on S.

On a Drygas inequality

ABSTRACT The goal of the paper is to give sufficient conditions for which the solution to the Drygas inequality is continuous.

On the lower hull of convex functions

SummaryLet (X, ℱ) be a topological space. For any functionf: D→[− ∞, ∞) (whereD ⊂ X), thelower hull mf:D →[− ∞, ∞) off is defined by

On separation theorems for subadditive and superadditive functionals

m_f (x) = m_{f\left| T \right.} (x) = \mathop {\sup \inf }\limits_{U \in T_x \in U \cap D} f(t),x \in D,

Some remarks on subquadratic functions

where ℱx denotes the family of all open sets containing x. The main result of the paper is that, ifX is a real linear topological Baire space,D ⊂ X is convex and open, andf: D→[− ∞, ∞) isJ-convex, then the functionmf is convex and continuous. (In the case of a single real variable this result goes back to F. Bernstein and G. Doetsch, 1915.)Now letX be a real linear space. A setG ⊂ X is calledalgebraically open if for everyx ∈ G andy ∈ X there exists anε = ε(x, y) > 0 such thatx + λy ∈ G for λ ∈(−ε, ε). The family ℱ (X) of all algebraically open subsets ofX is a topology inX, which, however, is not linear (unless dimX = 1). For any functionf: D →[− ∞, ∞) thealgebraic lower hull mf*:D →[− ∞, ∞) is defined asmf* =mf|ℱ(x). Again, ifD is convex and open andf isJ-convex, then the functionmf* is convex and continuous with respect to the topology ℱ(X). IfX is a real linear topological space,D ⊂ X is convex and open, andf: D →[− ∞, ∞) is an arbitrary function, then bothmf andmf* are well defined inD. We always havemf⩽ mf*⩽ f; moreover,mf* =f wheneverf is convex, andmf* =mf wheneverf isJ-convex and dimX is finite, but in general neither of these equalities holds.A number of related questions are also discussed. In particular, it is shown that, ifX is a real linear topological space,D ⊂ X is convex and open, andf: D →[− ∞, ∞) is aJ-convex function which is lower semicontinuous at every point of a setS ⊂ D containing a second category Baire subset, thenf is convex and continuous.