By Marek Kuczma
Marek Kuczma was once born in 1935 in Katowice, Poland, and died there in 1991.
After completing highschool in his domestic city, he studied on the Jagiellonian college in Kraków. He defended his doctoral dissertation lower than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he got a place on the Katowice department of the Jagiellonian college (now collage of Silesia, Katowice), and labored there until his death.
Besides his a number of administrative positions and his notable instructing task, he finished first-class and wealthy clinical paintings publishing 3 monographs and one hundred eighty medical papers.
He is taken into account to be the founding father of the prestigious Polish university of useful equations and inequalities.
"The moment 1/2 the name of this ebook describes its contents accurately. most likely even the main committed professional don't have concept that approximately three hundred pages could be written almost about the Cauchy equation (and on a few heavily similar equations and inequalities). And the publication is on no account chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree conception, topology and algebra. half II supplies information on options of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex capabilities, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex features of upper order, subadditive services and balance theorems). It concludes with an expedition into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)
"This publication is a true vacation for the entire mathematicians independently in their strict speciality. you can still think what deliciousness represents this publication for useful equationists." (B. Crstici, Zentralblatt für Mathematik)
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Extra resources for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality
18) The sets F ∩ An n ∈ N, are pairwise disjoint measurable sets. Since F ⊂ F ⊂F∩ ∞ An = n=1 ∞ ∞ An , also n=1 (F ∩ An ). Moreover, we have F ∩ An ⊂ A ∩ An , n ∈ N. Hence n=1 by the monotonicity of the inner measure ∞ m(F ) m ∞ (F ∩ An ) = n=1 ∞ m(F ∩ An ) = n=1 ∞ mi (F ∩ An ) n=1 mi (A ∩ An ) . 19) imply ∞ mi (A ∩ An ) .
3. Let Y be a separable topological space. If the set A ⊂ X × Y has the Baire property, then there exists a set Q ⊂ X, of the ﬁrst category, such that for every x ∈ Q = X \ Q the set A[x] has the Baire property. Proof. We have A = (G ∪ P ) \ R, where G is open, and P, R are of the ﬁrst category. 7 there exist sets Q1 ⊂ X and Q2 ⊂ X such that for every x ∈ Q1 the set P [x] is of the ﬁrst category, and for every x ∈ Q2 the set R[x] is of the ﬁrst category. Put Q = Q1 ∪ Q2 . For x ∈ Q the sets P [x] and R[x] both are of the ﬁrst category (in Y ), and Q itself is of the ﬁrst category (in X).
A is perfect. , B • ∈ K and B • ⊂ A. By the deﬁnition of B we have B∩B • = ∅ and B = B\B • . 1 card B = card (B\B • ) ℵ0 . Any subset D of X may be considered as a space, so D = A∪B with A = Ad and card B ℵ0 . But in the formula A = Ad the accumulation points of A are taken with respect to the relative topology of D. (These are points x such that card (D∩U ∩A) > 1 for every neighbourhood U of x). But since every accumulation point of A in D is an accumulation point of A in X card (D ∩ U ∩ A) > 1 implies card (U ∩ A) > 1 , Ad in D is contained in Ad in X.