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Selecting basic sequences in φ-stable Banach spaces

Tadeusz Figiel, Ryszard Frankiewicz, Ryszard A. Komorowski, Czesław Ryll-Nardzewski (2003)

Studia Mathematica

In this paper we make use of a new concept of φ-stability for Banach spaces, where φ is a function. If a Banach space X and the function φ satisfy some natural conditions, then X is saturated with subspaces that are φ-stable (cf. Lemma 2.1 and Corollary 7.8). In a φ-stable Banach space one can easily construct basic sequences which have a property P(φ) defined in terms of φ (cf. Theorem 4.5). This leads us, for appropriate functions φ, to new results on the existence of unconditional...

Separability of Real Normed Spaces and Its Basic Properties

Kazuhisa Nakasho, Noboru Endou (2015)

Formalized Mathematics

In this article, the separability of real normed spaces and its properties are mainly formalized. In the first section, it is proved that a real normed subspace is separable if it is generated by a countable subset. We used here the fact that the rational numbers form a dense subset of the real numbers. In the second section, the basic properties of the separable normed spaces are discussed. It is applied to isomorphic spaces via bounded linear operators and double dual spaces. In the last section,...

Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven (2005)

Colloquium Mathematicae

We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xₙ) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence ( x n j ) of (xₙ) such that i n f j ≠ k | | x - ( x n j - x n k ) | | ≥ 1 + δ X ( 2 / 3 ε ) , where δ X is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space contains...

Separating polynomials on Banach spaces.

R. Gonzalo, J. A. Jaramillo (1997)

Extracta Mathematicae

In this paper we survey some recent results concerning separating polynomials on real Banach spaces. By this we mean a polynomial which separates the origin from the unit sphere of the space, thus providing an analog of the separating quadratic form on Hilbert space.

Sequentially Right Banach spaces of order p

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order p , and those defined by the dual property, the sequentially Right * Banach spaces of order p for 1 ≤ p ≤ ∞ . These classes of Banach spaces are characterized by the notions of L p -limited sets in the corresponding dual space and R p * subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space X and a reflexive Banach space...

Singlevaluedness of monotone operators on subspaces of GSG spaces

Martin Heisler (1996)

Commentationes Mathematicae Universitatis Carolinae

We extend Zajíček’s theorem from [Za] about points of singlevaluedness of monotone operators on Asplund spaces. Namely we prove that every monotone operator on a subspace of a Banach space containing densely a continuous image of an Asplund space (these spaces are called GSG spaces) is singlevalued on the whole space except a σ -cone supported set.

Smith's counterexample about uniform rotundity in every direction.

Manuel Fernández, Isidro Palacios (2000)

Extracta Mathematicae

It is an open question when the direct sum of normed spaces inherits uniform rotundity in every direction from the factor spaces. M. Smith [4] showed that, in general, the answer is negative. The purpose of this paper is to carry out a complete study of Smith's counterexample.

Smooth approximations without critical points

Petr Hájek, Michal Johanis (2003)

Open Mathematics

In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set...

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