Maite Fernández-Unzueta

Extension of polynomials defined on subspaces

Let k ∈ N and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace F ⊂ E, there exists a continuous morphism φk,F : P( k F) → P( k E) satisfying φk,F (P)|F = P, if and only E is isomorphic to a Hilbert space.

The Segre cone of Banach spaces and multilinear mappings

ABSTRACT We prove that any pair of reasonable cross norms defined on the tensor product of n Banach spaces induce -Lipschitz equivalent metrics (thus, a unique topology) on the set of vectors of rank . With this, we define the Segre cone of Banach spaces, , and state when is closed. To study multilinear mappings, we introduce Σ-operators as mappings on naturally associated with them. We prove that a multilinear mapping is bounded if and only if its associated Σ-operator is Lipschitz on the Segre cone.