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

Besides his numerous administrative positions and his striking educating task, he complete very good 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 tuition of practical equations and inequalities.

"The moment half the identify of this publication describes its contents effectively. most likely even the main committed professional don't have idea that approximately three hundred pages may be written with regards to the Cauchy equation (and on a few heavily similar equations and inequalities). And the publication is under no circumstances chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree concept, topology and algebra. half II offers information on strategies of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex capabilities, 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 capabilities of upper order, subadditive services and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This publication is a true vacation for all of the mathematicians independently in their strict speciality. you will think what deliciousness represents this publication for useful equationists." (B. Crstici, Zentralblatt für Mathematik)

 

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

Example text

Nm 1 2m . 4. nm ∩ cl Kn(m) , nm+1 ∈ N . nm } , m, n1 , . . nm , m, n1 , . . , nm , nm+1 ∈ N. nm . 9) defines a function f : z → X. nm , n1 . . nm ∈ N, form a cover of X. nm . The sequence {ni } represents a point z ∈ z and we have f (z) = x. Thus f is onto: f (z) = X . It remains to show that f is continuous. Fix an ε > 0 and choose a p ∈ N so that 1 1 < ε. Put δ = p+1 . Take z , z ∈ z, z = {ni }, z = {ni }. (z , z ) < δ means 2p−1 2 ∞ i=1 1 1 |ni − ni | < p+1 . 10) Since ni and ni are integers, ni = ni implies |ni − ni | 1 + |ni − ni | 1 .

Thus we have αn < α < Ω for every n ∈ N. 1 (i) Aαn ⊂ Mα for every n ∈ N, whence An ∈ Mα for every n ∈ N. Hence ∞ An ∈ (Mα )σ ⊂ n=1 Mξ ξ<α+1 α<Ω Mα = α<Ω = Aα+1 ⊂ Aα . α<Ω Aα . Then A ∈ Aβ for a certain β < Ω. 1 Now take a set A ∈ (ii) A ∈ Mβ ⊂ σ Aα . Consequently α<Ω Aα is a σ-algebra. 3. B(X) is the smallest class K of subsets of X with the properties: (i) F (X) ⊂ K, (ii) if An ∈ K for n ∈ N, then ∞ An ∈ K and n=1 ∞ An ∈ K. n=1 Proof. Clearly B(X) has properties (i) and (ii), so K ⊂ B(X) . To prove the converse inclusion, we will show that for every α < Ω Aα ∪ Mα ⊂ K .

Hence also card B ℵ0 , and consequently α ℵ0 . , α < Ω. Thus we have αn < α < Ω for every n ∈ N. 1 (i) Aαn ⊂ Mα for every n ∈ N, whence An ∈ Mα for every n ∈ N. Hence ∞ An ∈ (Mα )σ ⊂ n=1 Mξ ξ<α+1 α<Ω Mα = α<Ω = Aα+1 ⊂ Aα . α<Ω Aα . Then A ∈ Aβ for a certain β < Ω. 1 Now take a set A ∈ (ii) A ∈ Mβ ⊂ σ Aα . Consequently α<Ω Aα is a σ-algebra. 3. B(X) is the smallest class K of subsets of X with the properties: (i) F (X) ⊂ K, (ii) if An ∈ K for n ∈ N, then ∞ An ∈ K and n=1 ∞ An ∈ K. n=1 Proof. Clearly B(X) has properties (i) and (ii), so K ⊂ B(X) .

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