By Marek Kuczma

ISBN-10: 3764387483

ISBN-13: 9783764387488

Marek Kuczma used to be born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian collage in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the 12 months of his habilitation, in 1963, he acquired a place on the Katowice department of the Jagiellonian college (now college of Silesia, Katowice), and labored there until eventually his death.

Besides his numerous administrative positions and his awesome instructing task, he comprehensive first-class and wealthy clinical paintings publishing 3 monographs and a hundred and eighty clinical papers.

He is taken into account to be the founding father of the prestigious Polish institution of practical equations and inequalities.

"The moment half the name of this publication describes its contents properly. most likely even the main dedicated professional shouldn't have concept that approximately three hundred pages could be written with regards to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the publication is in no way chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree thought, topology and algebra. half II offers information on recommendations 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 offers with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex services of upper order, subadditive features 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 possibly can think what deliciousness represents this e-book for useful equationists." (B. Crstici, Zentralblatt für Mathematik)

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**Additional resources for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality**

**Example text**

2) B \ A ∈ Σ. Since A ⊂ B, we have A = B \ (B \ A) ∈ Σ. 2. Every analytic set has the Baire property. In Part II of the present book we will encounter many sets without the Baire property, and hence non-analytic. 8. 8 Cantor-Bendixson theorem The Cantor-Bendixson theorem belongs to standard part of any university course on the topology of metric spaces. But we prove it here because of its application in the theory of analytic sets. Let X be a metric space and A ⊂ X. A point x ∈ X is called a point of accumulation of A iﬀ for every neighbourhood U of x we have card (U ∩ A) > 1.

7. Let Y be a separable topological space, and let A ⊂ X × Y be a set of the ﬁrst category. 5) is of the ﬁrst category. Proof. We have A = ∞ An , where the sets An ⊂ X × Y are nowhere dense. 2, for every n ∈ N, there exists a set Pn ⊂ X, of the ﬁrst category, such that for every x ∈ Pn the set An [x] is nowhere dense. Put ∞ P = Pn . 8) n=1 is of the ﬁrst category. 8) is true observe that y ∈ A[x] ⇔ (x, y) ∈ A ⇔ there exists an n ∈ N such that (x, y) ∈ An ⇔ there exists an n ∈ N such that y ∈ An [x] ⇔ y ∈ ∞ n=1 An [x].

We have Aα = α<Ω Mα = B(X) . α<Ω Proof. 1 (i) Aα ⊂ Mα+1 ⊂ Mα , and Aα ⊂ Mα . Similarly, for every α < Ω we have α<Ω α<Ω Mα ⊂ Aα+1 ⊂ α<Ω Aα , and α<Ω Mα ⊂ α<Ω Aα . 1 (iii) α<Ω Mα , α<Ω Mα ⊂ B(X). , it must contain B(X). Take a sequence of sets An ∈ Aα . Then, for every n ∈ N, there exists an α<Ω αn < Ω such that An ∈ Aαn . 4 there exists an ordinal number α greater than every number αn . This has been constructed as α = B + 1, where, in the present case B = ∞ Γ(αn ). We have for every n ∈ N , Γ(αn ) = αn < Ω, whence n=1 card Γ(αn) = αn ℵ0 .

### An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality by Marek Kuczma

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