Read e-book online An Introduction to the Theory of Functional Equations and PDF

By Marek Kuczma (auth.), Attila Gilányi (eds.)

ISBN-10: 3764387483

ISBN-13: 9783764387488

ISBN-10: 3764387491

ISBN-13: 9783764387495

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 college in Kraków. He defended his doctoral dissertation below the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he bought a place on the Katowice department of the Jagiellonian college (now collage of Silesia, Katowice), and labored there until eventually his death.

Besides his numerous administrative positions and his impressive educating job, he finished 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 institution of sensible equations and inequalities.

"The moment half the identify of this booklet describes its contents properly. most likely even the main committed expert do not need notion that approximately three hundred pages could be written on the subject of the Cauchy equation (and on a few heavily comparable equations and inequalities). And the publication is not at all chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree idea, topology and algebra. half II supplies info on strategies of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III offers with similar 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 ebook is a true vacation for the entire mathematicians independently in their strict speciality. you can actually think what deliciousness represents this booklet for practical equationists." (B. Crstici, Zentralblatt für Mathematik)

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

Example text

Y1 , y2 , x3 , . } , {y1 , y2 , y3 , . 5) ............... , yi ∈ Yi . The cardinality of Y equals that of ∞ ∞ × Yi , and thus Y is countable. Take n=1 i=1 an x = {xi } ∈ X. 6) lim yni = xi . 1 that lim yn = x . n→∞ Thus Y is dense in X, and consequently X is separable. 2. If all spaces Xi are complete, then so is also X. Proof. Let {xn } ⊂ X , xn = {xni } , be an arbitrary Cauchy sequence in X. 1) (xn , xn+k ) 1 i (xni , xn+k,i ) 2i 1 + i (xni , xn+k,i) 30 Chapter 2. Topology for all n , k , i ∈ N, whence it follows that for every i ∈ N the sequence {xni } ⊂ Xi is a Cauchy sequence in Xi .

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 .

Piccard 43 Proof. X itself is of the second category, since it contains sets of the second category. Then every non-empty open subset of X is of the second category. In fact, suppose that G ⊂ X is non-empty, open and of the first category. Fix an a ∈ G. 5) are homeomorphisms and x ∈ G − a + x. Thus X is of the first category at every point x ∈ X. 4 X is of the first category, which is impossible. , there exist non-empty open sets G, H (if G, H were empty, the sets A, B would be of the first category) and first category sets P , Q , R , S such that A = (G ∪ P ) \ R , B = (H ∪ Q) \ S .

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An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality by Marek Kuczma (auth.), Attila Gilányi (eds.)


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