Currently displaying 1 – 12 of 12

Showing per page

Order by Relevance | Title | Year of publication

On the volume method in the study of Auerbach bases of finite-dimensional normed spaces

Anatolij Plichko — 1996

Colloquium Mathematicae

In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr MykhaylyukAnatolij Plichko — 2015

Colloquium Mathematicae

We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function F y ( x ) : = f y ' ( x , y ) has finite variation, then almost everywhere on the rectangle the partial derivative f y x ' ' exists. We construct a separately twice differentiable function whose partial derivative f x ' is discontinuous with respect to the second variable on a set of positive...

The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.

Taras BanakhAnatolij Plichko — 2006

RACSAM

Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...

Decomposition of Banach Space into a Direct Sum of Separable and Reflexive Subspaces and Borel Maps

Plichko, Anatolij — 1997

Serdica Mathematical Journal

* This paper was supported in part by the Bulgarian Ministry of Education, Science and Technologies under contract MM-506/95. The main results of the paper are: Theorem 1. Let a Banach space E be decomposed into a direct sum of separable and reflexive subspaces. Then for every Hausdorff locally convex topological vector space Z and for every linear continuous bijective operator T : E → Z, the inverse T^(−1) is a Borel map. Theorem 2. Let us assume the continuum hypothesis. If a Banach...

On bases in Banach spaces

We investigate various kinds of bases in infinite-dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in as well as in separable Banach spaces.

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

On a functional-analysis approach to orthogonal sequences problems.

Sea T un operador lineal acotado e inyectivo de un espacio de Banach X en un espacio de Hilbert H con rango denso y sea {x} ⊂ X una sucesión tal que {Tx} es ortogonal. Se estudian propiedades de {Tx} dependientes de propiedades de {x}. También se estudia la ""situación opuesta"", es decir, la acción de un operador T : H → X sobre sucesiones ortogonales.

Page 1

Download Results (CSV)