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Deviation from weak Banach–Saks property for countable direct sums

Andrzej Kryczka (2015)

Annales UMCS, Mathematica

We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν) is equal to the supremum of such deviations attained on the coordinates Xν. This is a quantitative version for operators of the result...

Diagonals of projective tensor products and orthogonally additive polynomials

Qingying Bu, Gerard Buskes (2014)

Studia Mathematica

Let E be a Banach space with 1-unconditional basis. Denote by Δ ( ⊗ ̂ n , π E ) (resp. Δ ( ⊗ ̂ n , s , π E ) ) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by Δ ( ⊗ ̂ n , | π | E ) (resp. Δ ( ⊗ ̂ n , s , | π | E ) ) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to E [ n ] , the completion of the n-concavification of...

Diameter 2 properties and convexity

Trond Arnold Abrahamsen, Petr Hájek, Olav Nygaard, Jarno Talponen, Stanimir Troyanski (2016)

Studia Mathematica

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of C[0,1] with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.

Diameter, extreme points and topology

J. C. Navarro-Pascual, M. G. Sanchez-Lirola (2009)

Studia Mathematica

We study the extremal structure of Banach spaces of continuous functions with the diameter norm.

Dichotomies for 𝐂 0 ( X ) and 𝐂 b ( X ) spaces

Szymon Głąb, Filip Strobin (2013)

Czechoslovak Mathematical Journal

Jachymski showed that the set ( x , y ) ∈ 𝐜 0 × 𝐜 0 : ∑ i = 1 n α ( i ) x ( i ) y ( i ) n = 1 ∞ is bounded is either a meager subset of 𝐜 0 × 𝐜 0 or is equal to 𝐜 0 × 𝐜 0 . In the paper we generalize this result by considering more general spaces than 𝐜 0 , namely 𝐂 0 ( X ) , the space of all continuous functions which vanish at infinity, and 𝐂 b ( X ) , the space of all continuous bounded functions. Moreover, we replace the meagerness by σ -porosity.

Diophantine approximation in Banach spaces

Lior Fishman, David Simmons, Mariusz Urbański (2014)

Journal de Théorie des Nombres de Bordeaux

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

Direct limit of matricially Riesz normed spaces

J. V. Ramani, Anil Kumar Karn, Sunil Yadav (2006)

Commentationes Mathematicae Universitatis Carolinae

In this paper, the ℱ -Riesz norm for ordered ℱ -bimodules is introduced and characterized in terms of order theoretic and geometric concepts. Using this notion, ℱ -Riesz normed bimodules are introduced and characterized as the inductive limits of matricially Riesz normed spaces.

Directional moduli of rotundity and smoothness

Michael O. Bartlett, John R. Giles, Jon D. Vanderwerff (1999)

Commentationes Mathematicae Universitatis Carolinae

We study various notions of directional moduli of rotundity and when such moduli of rotundity of power type imply the underlying space is superreflexive. Duality with directional moduli of smoothness and some applications are also discussed.

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