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 below the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he received a place on the Katowice department of the Jagiellonian collage (now college of Silesia, Katowice), and labored there until eventually his death.
Besides his a number of administrative positions and his extraordinary educating job, he complete first-class and wealthy clinical paintings publishing 3 monographs and one hundred eighty clinical papers.
He is taken into account to be the founding father of the distinguished Polish university of sensible equations and inequalities.
"The moment half the identify of this booklet describes its contents appropriately. most likely even the main committed professional should not have idea that approximately three hundred pages will be written almost about the Cauchy equation (and on a few heavily comparable equations and inequalities). And the booklet is under no circumstances chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree conception, topology and algebra. half II provides information on suggestions of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex capabilities, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with similar equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex features of upper order, subadditive capabilities and balance theorems). It concludes with an day trip into the sphere of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)
"This e-book is a true vacation for all of the mathematicians independently in their strict speciality. you can still think what deliciousness represents this booklet for practical 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
Borel sets 25 Now let Y be a separable topological space. 5). 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.
Then cf. 3) B \ Z = D(A) \ Z ∪ (C \ Z) ⊂ D(Z) \ Z ∪ (C \ Z) ⊂ D(Z) \ Z ∪ C is of the ﬁrst category (cf. 5). 6 The assumption that X is separable is not essential and can be omitted. Cf. the footnote 4 on p. 21. 3. Borel sets 25 Now let Y be a separable topological space. 5). 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.
4) the set A is of the ﬁrst category. 5. For every set A ⊂ X, the set D(A) is closed. Proof. We shall show that the set X \ D(A) is open. Take an x ∈ D(A). Then there exists an open neighbourhood Ux of x such that A ∩ Ux is of the ﬁrst category. Now, since Ux is a neighbourhood of every point from Ux , we have Ux ⊂ X \ D(A). Consequently x is an inner point of X \ D(A), which shows that X \ D(A) is open, and so D(A) is closed. 6. If the space X is separable, then for every set A ⊂ X, the set A\D(A) is of the ﬁrst category.
An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality by Marek Kuczma