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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 -Riesz norm for ordered -bimodules is introduced and characterized in terms of order theoretic and geometric concepts. Using this notion, -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.

Directionally Euclidean structures of Banach spaces

Jarno Talponen (2011)

Studia Mathematica

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. ...

Disjointification of martingale differences and conditionally independent random variables with some applications

Sergey Astashkin, Fedor Sukochev, Chin Pin Wong (2011)

Studia Mathematica

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form f k ( s ) x k ( t ) k = 1 , where f k ’s are independent and xk’s are arbitrary random variables from a symmetric space X on [0,1]. The main results show that the form of these inequalities depends on which side of L₂ the space X lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with...

Distances between Hilbertian operator spaces

Seán Dineen, Cristina Radu (2014)

Studia Mathematica

We compute the completely bounded Banach-Mazur distance between different finite-dimensional homogeneous Hilbertian operator spaces.

Distances to convex sets

Antonio S. Granero, Marcos Sánchez (2007)

Studia Mathematica

If X is a Banach space and C a convex subset of X*, we investigate whether the distance d ̂ ( c o ¯ w * ( K ) , C ) : = s u p i n f | | k - c | | : c C : k c o ¯ w * ( K ) from c o ¯ w * ( K ) to C is M-controlled by the distance d̂(K,C) (that is, if d ̂ ( c o ¯ w * ( K ) , C ) M d ̂ ( K , C ) for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

Let E be a Banach space and let ( B E * ) and ( B E * ) denote the space of all Baire-one and affine Baire-one functions on the dual unit ball B E * , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between d i s t ( f , ( B E * ) ) and d i s t ( f , ( B E * ) ) , where f is an affine function on B E * . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

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