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