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Separable reduction theorems by the method of elementary submodels

Marek Cúth (2012)

Fundamenta Mathematicae

We simplify the presentation of the method of elementary submodels and we show that it can be used to simplify proofs of existing separable reduction theorems and to obtain new ones. Given a nonseparable Banach space X and either a subset A ⊂ X or a function f defined on X, we are able for certain properties to produce a separable subspace of X which determines whether A or f has the property in question. Such results are proved for properties of sets: of being dense, nowhere dense, meager, residual...

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.

Sequence spaces generated by moduli of smoothness.

J. Musielak, A. Waszak (1995)

Revista Matemática de la Universidad Complutense de Madrid

There are defined sequential moduli in the remainder form for real sequences. Properties of sequence spaces generated by means of the above moduli are investigated.

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