An Introduction to the Theory of Functional Equations and by Marek Kuczma

By Marek Kuczma

Marek Kuczma was once 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 bought a place on the Katowice department of the Jagiellonian college (now college of Silesia, Katowice), and labored there until his death.

Besides his a number of administrative positions and his awesome educating task, he entire very good and wealthy medical paintings publishing 3 monographs and one hundred eighty medical papers.

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

"The moment 1/2 the identify of this e-book describes its contents properly. most likely even the main dedicated expert don't have proposal that approximately three hundred pages may be written with regards to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the publication is in no way chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree thought, topology and algebra. half II provides info on suggestions of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with similar equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex features 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 booklet is a true vacation for all of the mathematicians independently in their strict speciality. you could think what deliciousness represents this booklet for practical equationists." (B. Crstici, Zentralblatt für Mathematik)

 

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Proof. 1) a set C ∈ Fσ , of the first category such that A \ D(A) ⊂ C. Put B = D(A) ∪ C. The set B has the Baire property, since both, D(A) (being closed; cf. 1), and evidently A = A∩D(A) ∪ A\D(A) ⊂ D(A)∪C = B. Let Z ⊃ A be an arbitrary set with the Baire property. Then cf. 3) B \ Z = D(A) \ Z ∪ (C \ Z) ⊂ D(Z) \ Z ∪ (C \ Z) ⊂ D(Z) \ Z ∪ C is of the first category (cf. 5). 6 The assumption that X is separable is not essential and can be omitted. Cf. the footnote 4 on p. 21. 3. Borel sets 25 Now let Y be a separable topological space.

Thus {xn } converges in F : lim xn = x ∈ F. 4. 2. Let X, Y be separable and complete metric spaces, A ⊂ X an analytic set, and g : A → Y a continuous function. Then the set g(A) ⊂ Y also is analytic. Proof. There exists a continuous function f : z → X such that f (z) = A. The function h = g ◦ f (the composite of g and f ) is continuous, h : z → Y , and h(z) = g(A). 3. The cartesian product of a finite or countable number of analytic sets is an analytic set. Proof. Let Ai , i ∈ I, where I = {1, .

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 .

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