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

Besides his a number of administrative positions and his impressive instructing task, 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 university of useful equations and inequalities.

"The moment 1/2 the identify of this booklet describes its contents correctly. most likely even the main committed professional shouldn't have concept that approximately three hundred pages should be written as regards to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the e-book is certainly not 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 info on ideas 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 bargains with similar equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex capabilities of upper order, subadditive capabilities and balance theorems). It concludes with an expedition into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This booklet is a true vacation for the entire mathematicians independently in their strict speciality. you'll be able to think what deliciousness represents this booklet 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**

1. If xn = {xni } ∈ X , n ∈ N, then the sequence {xn } converges: lim xn = x0 = {x0i } ∈ X n→∞ if and only if lim xni = x0i for every i ∈ N. 4. The space z 29 Proof. 1) (xn , x0 ) (xn , x0 ) = 0 . 4) for every i ∈ N. 4) holds for every i ∈ N. We have m (xn , x0 ) i=1 ∞ 1 1 i (xni , x0i ) + = 2i 1 + i (xni , x0i ) i=m+1 2i m i=1 1 1 i (xni , x0i ) + 2i 1 + i (xni , x0i ) 2m for every m ∈ N. 3). 1. If all spaces Xi are separable, then so is also X. Proof. For every i ∈ N there exists a countable set Yi ⊂ Xi , dense in Xi .

Hence it follows that the sets G \ A ⊂ R and B \ H ⊂ S are of the ﬁrst category. Fix elements g ∈ G and h ∈ H and put G0 = −g + G , H0 = H − h , A0 = −g + A , B0 = B − h. 5) are homeomorphisms the set G0 \ A0 = (−g + G) \ (−g + A) = −g + (G \ A) is of the ﬁrst category, and similarly the set H0 \ B0 = (H \ B) − h is of the ﬁrst category. Put U = G0 ∩ H0 . U is open, since so are G0 and H0 , and non-empty, since 0 ∈ U . Consequently U is of the second category. For every t ∈ U write Ut = U ∩ (t − U ) .

Iii) There exist sets H ∈ Gδ and B ∈ L such that A ⊂ H , A = H \ B , m(B) = 0 . (iv) For every ε > 0 there exist a closed set F and an open set V such that F ⊂ A , A \ F ⊂ V , m(V ) < ε . (v) There exist sets E ∈ Fσ and C ∈ L such that E ⊂ A , A = E ∪ C , m(C) = 0 . 1 will be our main tool in the sequel of this chapter. 48 Chapter 3. Measure Theory Let A ⊂ RN be an arbitrary set. We deﬁne the outer measure1 and the inner measure of A (denoted in the sequel by me (A) and mi (A), respectively) as me (A) = inf m(G), mi (A) = G⊃A G open sup m(F ) .

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

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