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Displaying similar documents to “Finite canonization”

Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

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We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that 2 ω ₁ is arbitrarily large,...

Compact operators and integral equations in the ℋ𝒦 space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space ℋ𝒦 of Henstock-Kurzweil integrable functions on [ a , b ] is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . In this paper, on each Fréchet space ℋ𝒦 ( X ) , an F -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the ℋ𝒦 ( X ) space. It is known that every control-convergent sequence in the ℋ𝒦 space always belongs to a ℋ𝒦 ( X ) space for some X . We illustrate how...

On Ozeki’s inequality for power sums

Horst Alzer (2000)

Czechoslovak Mathematical Journal

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Let p ∈ ( 0 , 1 ) be a real number and let n ≥ 2 be an even integer. We determine the largest value c n ( p ) such that the inequality ∑ i = 1 n | a i | p ≥ c n ( p ) holds for all real numbers a 1 , ... , a n which are pairwise distinct and satisfy min i ≠ j | a i - a j | = 1 . Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value c n ( p ) in the case p > 0 and n odd, and in the case p ≥ 1 and n even.

The topology of the space of ℋ𝒦 integrable functions in ℝ n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm ∥ · ∥ X is defined. A ∥ · ∥ X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a ∥ · ∥ X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every...

MAD families and P -points

Salvador García-Ferreira, Paul J. Szeptycki (2007)

Commentationes Mathematicae Universitatis Carolinae

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The Katětov ordering of two maximal almost disjoint (MAD) families 𝒜 and ℬ is defined as follows: We say that 𝒜 ≤ K ℬ if there is a function f : ω → ω such that f - 1 ( A ) ∈ ℐ ( ℬ ) for every A ∈ ℐ ( 𝒜 ) . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called K -uniform if for every X ∈ ℐ ( 𝒜 ) + , we have that 𝒜 | X ≤ K 𝒜 . We prove that CH implies that for every K -uniform MAD family 𝒜 there is a P -point p of ω * such that the set of all Rudin-Keisler predecessors of p is dense...