Marek Kuczma's An Introduction to the Theory of Functional Equations and PDF

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 lower 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 collage (now collage of Silesia, Katowice), and labored there until his death.

Besides his numerous administrative positions and his notable educating job, he comprehensive first-class and wealthy medical paintings publishing 3 monographs and a hundred and eighty medical papers.

He is taken into account to be the founding father of the distinguished Polish tuition of useful equations and inequalities.

"The moment half the name of this ebook describes its contents appropriately. most likely even the main committed expert wouldn't have suggestion that approximately three hundred pages should be written with regards to the Cauchy equation (and on a few heavily similar equations and inequalities). And the booklet is not at all chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree thought, topology and algebra. half II offers information on suggestions of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex features, 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 capabilities of upper order, subadditive features and balance theorems). It concludes with an expedition into the sphere 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. one could think what deliciousness represents this ebook for practical equationists." (B. Crstici, Zentralblatt für Mathematik)

 

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Extra info for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality

Example text

Suppose that for all β < α < Ω we have (∗∗) if Z ∈ Aβ , then Z ∈ Mβ , and conversely. Take a Z ∈ Aα . , for every n ∈ N there exists a ξn < α such ξ<α that En ∈ Mξn . Hence, by (∗∗), En ∈ Aξn , n ∈ N, and ∞ En ∈ Z = n=1 Aξ δ ξ<α The proof of the converse implication is analogous. = Mα . 3. Borel sets 27 (iii) This results from the fact that B(X) is a σ-algebra (again a proof by transfinite induction is necessary, which is left to the reader). 2. We have Aα = α<Ω Mα = B(X) . α<Ω Proof. 1 (i) Aα ⊂ Mα+1 ⊂ Mα , and Aα ⊂ Mα .

Prove the following version of the transfinite induction principle: Let (X, ) be a well-ordered set, x0 the smallest element in X, and Φ a propositional formula defined on X. If the following conditions are fulfilled (i) Φ (x0 ), (ii) If Φ (y) for y ∈ P (x), then Φ (x), x ∈ X, then Φ (x) for all x ∈ X. 13. Prove that if (A, ) is a well-ordered set similar to its subset A0 , then the similarity function f : A → A0 satisfies f (x) x for all x ∈ A. 14. Prove that in every family E of sets there exists a maximal subfamily E0 of mutually disjoint sets.

Prove that in every family E of sets there exists a maximal subfamily E0 of mutually disjoint sets. , X may be a metric space, or, in particular, RN . A set A ⊂ X is called nowhere dense iff int cl A = ∅. A set A ⊂ X is said to be of the first category iff A is a countable union of nowhere dense sets: ∞ Ai , int cl Ai = ∅, i ∈ N . 1) i=1 A set A ⊂ X which is not of the first category is said to be of the second category. A set A ⊂ X is called residual iff A = X \ A is of the first category. 1. If A ⊂ X is of the first category, then there exists a set B ∈ Fσ such that A ⊂ B, and B is of the first category.

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


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