# 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 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 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 received a place on the Katowice department of the Jagiellonian collage (now college of Silesia, Katowice), and labored there until eventually his death.

Besides his numerous administrative positions and his amazing educating job, he entire very good and wealthy clinical paintings publishing 3 monographs and one hundred eighty clinical papers.

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

"The moment 1/2 the name of this publication describes its contents correctly. most likely even the main committed expert do not need suggestion that approximately three hundred pages could be written almost about 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 necessary initial wisdom in set and degree conception, topology and algebra. half II offers information on strategies 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 capabilities of upper order, subadditive features and balance theorems). It concludes with an day trip 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 useful equationists." (B. Crstici, Zentralblatt für Mathematik)

**Read or Download An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality PDF**

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

**Sample text**

The set of all the accumulation points of A is denoted by Ad . A point x ∈ A is called a point of condensation of A iﬀ for every neighbourhood U of x we have card (U ∩ A) > ℵ0 . The set of all the condensation points of A is denoted by A• . A set A ⊂ X is called perfect iﬀ A = Ad . The reader will easily verify the following properties of the operations • and d : A• ⊂ Ad ⊂ cl A , (A ∪ B)• = A• ∪ B • , A ⊂ B ⇒ A• ⊂ B • , A• = cl A• , for every set collection K , Ad ⊂ A∈K A d , A∈K (A ∪ B)d = Ad ∪ B d , Add ⊂ Ad , cl A = A ∪ Ad .

7 there exist sets Q1 ⊂ X and Q2 ⊂ X such that for every x ∈ Q1 the set P [x] is of the ﬁrst category, and for every x ∈ Q2 the set R[x] is of the ﬁrst category. Put Q = Q1 ∪ Q2 . For x ∈ Q the sets P [x] and R[x] both are of the ﬁrst category (in Y ), and Q itself is of the ﬁrst category (in X). Now, we have A[x] = G[x] ∪ P [x] \ R[x] . For every x ∈ X the set G[x] is open (in Y ), therefore for every x ∈ Q the set A[x] has the Baire property. There are far reaching analogies between the sets of the ﬁrst category and the sets of Lebesgue measure zero (cf.

For every set A ⊂ X × Y and every x0 ∈ X we write A[x0 ] = {y ∈ Y | (x0 , y) ∈ A} . 2. Let Y be a separable5 topological space, and let A ⊂ X ×Y be a nowhere dense set. 5) is nowhere dense. , Kuratowski [196]). Since in the sequel the results of the present chapter will be used for X = RN , which is separable, we restrict ourselves to separable X only. Separable means here having an at most countable neighbourhood base. 8, the assumption that Y is separable is essential. 22 Chapter 2. Topology Proof.

### 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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