Vladimir I. Gurariy

Some results and open questions on spaceability in function spaces

A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y subset of M boolean OR {0}. In this article we prove that, for every infinite dimensional closed subspace X of C[0, 1], the set of functions in X having infinitely many zeros in [0, 1] is spaceable in X. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0, 1] or Muntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0, 1], as well as oscillating and annulling properties of subspaces of C[0, 1].

Geometry of Müntz spaces and related questions

Preface.- Part I Subspaces and Sequences in Banach Spaces: Disposition of Subspaces.- Sequences in Normed Spaces.- Isomorphism, Isometries and Embeddings.- Spaces of Universal Disposition.- Bounded Approximation Properties.- Part II On the Geometry of Muntz Sequences: Coefficient Estimates and the Muntz Theorem.- Classification and Elementary Properties of Muntz Sequences.- More on the Geometry of Muntz Sequences and Muntz Polynomials.- Operators of Finite Rank and Bases in Muntz Spaces.- Projection Types and the Isomorphism Problem for Muntz Spaces.- The Classes [M], A, P, and Pe.- Finite Dimensional Muntz Limiting Spaces in C.- References.- Index.

On Moduli of Convexity in Banach Spaces

We provide an example to show that the moduli of convexity δE and βE are different. Our example allows us to prove that 2 is the best possible constant in the inequality δE ≤ β E ≤ 2 δ E and that βE does not need to be a convex function.

Spaces of Universal Disposition

A Banach space U is called universal (for all separable Banach spaces) if for each separable Banach space X there is a subspace Y in U such that X is isometric to Y .

Bounded Approximation Properties

Throughout this section let X be a separable Banach space. We investigate certain approximation properties for X and deal with the question under which additional condition then X has a basis. In this context we also give a sufficient criterion for X to be a dual Banach space.

Isomorphisms, Isometries and Embeddings

Problems of isomorphisms and isometries play a central role in the theory of normed spaces. While two isomorphic normed spaces are identical from the linear point of view two isometric normed spaces are identical from a geometric point if view.

The Classes [ M ], A , P and P ε

In this chapter we study more general classes of subspaces of C[0, 1] which contain the class of Muntz spaces and we discuss theorems on compactness (11.1.2 and 11.2.6) as well as on interpolation (11.3.1). Moreover, we investigate which of these spaces can be embedded into c0. Finally we deal with notions of universality for Muntz sequences and spaces.

Disposition of Subspaces

In this chapter we discuss how two or more subspaces in a Banach space affect each other by their position in a Banach space and we give applications in the geometry of Banach spaces.

More on the Geometry of Müntz Sequences and Müntz Polynomials

In this chapter we continue the analysis of Muntz polynomials f(t) = Σk=1nαktλk where 0 < λ1 < λ2 <.

Sequences in Normed Spaces

Here we study, among other things, completeness and minimality of sequences in Banach spaces. This includes a discussion of bases and finite dimensional Schauder decompositions. Moreover this gives rise to the introduction of uniformly convex and uniformly smooth Banach spaces.