In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on ...

In the spirit of the work of Grothendieck, we introduce and study natural symmetric n-fold tensor norms. These are norms obtained from the projective norm by some natural operations. We prove that there are exactly six ...

We study tensor norms that destroy unconditionality in the following sense: for every Banach space E with unconditional basis, the n-fold tensor product of E (with the corresponding tensor norm) does not have unconditional ...

We give the symmetric version of five lemmas which are essential for the theory of tensor products (and norms). These are: the approximation, extension, embedding, density and local technique lemma. Some applications of ...

We introduce the symmetric-Radon-Nikodým property (sRN pr operty) for finitely generated s-tensor norms β of order n and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β is a projective ...

We study whether the operator space V ∗∗ α ⊗ W∗∗ can be identified with a subspace of the bidual space (V α ⊗ W) ∗∗, for a given operator space tensor norm. We prove that this can be done if α is finitely generated and V ...

We show that for every infinite-dimensional normed space E and every k ≥ 3 there are extendible k-homogeneous polynomials which are not integral. As a consequence, we prove a symmetric version of a result of John.