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Giving a partial answer to a conjecture formulated by Aron, Boyd, Ryan and Zalduendo, we show that if a real Banach space X is not linearly and continuously injected into a Hilbert space, then for any 2-homogeneous continuous polynomial P on X, its zero-set is not separable. For this class of spaces, we also prove that, if P is semidefinite, then is a nonseparable linear subspace. [ABSTRACT FROM AUTHOR]

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 ⊂ M ∪ {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 Müntz 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]. [ABSTRACT FROM AUTHOR]

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed sub-spaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. [ABSTRACT FROM AUTHOR]

Abstract: A classical inequality due to Bohnenblust and Hille states that for every positive integer m there is a constant so that for every positive integer N and every m-linear mapping , where . The value of was improved to by S. Kaijser and more recently H. Quéffelec and A. Defant and P. Sevilla-Peris remarked that also works. The Bohnenblust–Hille inequality also holds for real Banach spaces with the constants . In this note we show that a recent new proof of the Bohnenblust–Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for for all values of . In particular, we will also show that, for real scalars, if m is even with , then We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants. [Copyright &y& Elsevier]

Abstract: For each pair of numbers with , we consider the norm on given by for every . We investigate some geometrical properties of these norms. We provide an explicit formula for , a full description of the extreme points of the corresponding unit balls and a parametrization and a plot of their unit spheres. [Copyright &y& Elsevier]

We show that there exists an infinite dimensional vector space every non–zero element of which is a non–measurable function. Moreover, this vector space can be chosen to be closed and to have dimension β for any cardinality β. Some techniques involving measure theory and density characters of Banach spaces are used. [ABSTRACT FROM AUTHOR]

In this note, the linear structure of the family H e ( G ) of holomorphic functions in a domain G of a complex Banach space that are not holomorphically continuable beyond the boundary of G is analyzed. More particularly, we prove that H e ( G ) contains, except for zero, a closed (and a dense) vector space having maximal dimension, as well as a maximally generated free algebra. The results obtained complete a number of previous ones by several authors. [ABSTRACT FROM AUTHOR]