Hahn–Banach extension of multilinear forms and summability

Abstract

The aim of this paper is to investigate close relations between the validity of Hahn-Banach extension theorems for multilinear forms on Banach spaces and summability properties of sequences from these spaces. A case of particular importance occurs when we consider Banach spaces which have the property that every bilinear form extends to any superspace.

Representation of Extendible Bilinear Forms

“…Corollary 13 is already known (see, for example, Proposition 3.1 of [12]); although the current author has not seen it demonstrated using the approach given here.…”

“…The case considered here can be proved with similar arguments.) Therefore, by Proposition 7, it suffices to prove θ µ dν = θ µ dν for countably valued bounded measurable functions f and g in (12).…”

“…Since then, several authors have found various Hahn-Banach extension theorems, such as in [5,12] and more recently in [4]. (See also the references therein.…”

Composition results for strongly summing and dominated multilinear operators

“…Without any claim of completeness we mention: absolutely summing multilinear operators, see [1,2,9,24,37]; multiple summing multilinear operators, see [3,8,28,32]; dominated multilinear operators, see [21,27,30,31,33]. For the theory of polynomials in Banach spaces and their applications the interested reader can consult [17,18,29] and for the connection between holomorphy types and ideals of multilinear mappings we recommend the reader [7].…”

Dominated polynomials on infinite dimensional spaces

“…The following results were obtained with the help of the theory of absolutely summing multilinear mappings: (i) for every n ∈ N, a tensor norm of order n constructed by Pérez-García and Villanueva [41] is shown in Defant and Pérez-García [20] to preserve unconditionality for L p -spaces; (ii) Defant et al [19] provides optimal estimates for the width of Bohr's strip for Dirichlet series in infinite dimensional Banach spaces; (iii) applications to quantum information theory are obtained by Pérez-García et al [42], where it is proved that, contrary to the bipartite case, tripartite Bell inequalities can be unboundedly violated; (iv) in Jarchow et al [27], the existence of Hahn-Banach-type extension theorems for multilinear forms is strongly connected to the structural properties of the underlying spaces.…”