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 college in Krak?w. He defended his doctoral dissertation below the supervision of Stanislaw Golab. within the yr 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 impressive educating task, he finished first-class and wealthy medical paintings publishing 3 monographs and one hundred eighty clinical papers.

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

"The moment 1/2 the name of this booklet describes its contents thoroughly. most likely even the main dedicated expert wouldn't have concept that approximately three hundred pages will be written as regards to the Cauchy equation (and on a few heavily similar equations and inequalities). And the publication is not at all chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree conception, topology and algebra. half II supplies info 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 bargains with similar equations and inequalities (in specific, Pexider, Hossz?, and conditional equations, derivations, convex services of upper order, subadditive capabilities 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. you possibly can think what deliciousness represents this ebook for useful equationists." (B. Crstici, Zentralblatt f?r Mathematik)

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**Example text**

We must distinguish two cases. 1. α = β + 1. Then Mβ ⊂ (Mβ )σ ⊂ Mξ = Aα . σ ξ<α Further, Aξ ⊂ ξ<β Aξ , ξ<α whence also Mβ = Aξ δ ⊂ Aξ ξ<β δ = Mα . ξ<α Hence Mβ ⊂ Aα ∩ Mα . Similarly it is proved that Aβ ⊂ Aα ∩ Mα . Thus (i) holds. 2. α > β + 1. For the inductive proof assume that (∗) for every γ < α, if η < γ, then Aη ∪ Mη ⊂ Aγ ∩ Mγ . Put γ = β + 1. Then β < γ < α. By (∗) Aβ ∪ Mβ ⊂ Aγ ∩ Mγ . Hence Aβ ∪ Mβ ⊂ Aξ ∩ ξ<α Mξ ⊂ Aξ ξ<α δ ∩ ξ<α Mξ σ = Mα ∩ Aα . ξ<α Thus (i) holds in this case, too. (ii) The proof is again by transﬁnite induction with respect to α.

Nm ∈ Σ , m, n1 , . . nm } . 38 Chapter 2. Topology For the set A choose a set B ∈ Σ such that A ⊂ B and if A ⊂ Z ∈ Σ, then every Y ⊂ B \ Z belongs to Σ. nm , k1 , . . kj ∈ Σ according to the same pattern. nm , n1 , . . nm . nm = B. kj l ∈ Σ , l=1 since Σ is a σ-algebra. kj l ∈ Σ . nm \ z m=0 Let [. nm l . nm l . Then ∞ [. nm \ ∞ ∞ [. ] = m=0 z [. ] . 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.

We must distinguish two cases. 1. α = β + 1. Then Mβ ⊂ (Mβ )σ ⊂ Mξ = Aα . σ ξ<α Further, Aξ ⊂ ξ<β Aξ , ξ<α whence also Mβ = Aξ δ ⊂ Aξ ξ<β δ = Mα . ξ<α Hence Mβ ⊂ Aα ∩ Mα . Similarly it is proved that Aβ ⊂ Aα ∩ Mα . Thus (i) holds. 2. α > β + 1. For the inductive proof assume that (∗) for every γ < α, if η < γ, then Aη ∪ Mη ⊂ Aγ ∩ Mγ . Put γ = β + 1. Then β < γ < α. By (∗) Aβ ∪ Mβ ⊂ Aγ ∩ Mγ . Hence Aβ ∪ Mβ ⊂ Aξ ∩ ξ<α Mξ ⊂ Aξ ξ<α δ ∩ ξ<α Mξ σ = Mα ∩ Aα . ξ<α Thus (i) holds in this case, too. (ii) The proof is again by transﬁnite induction with respect to α.

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