Series representation in tensor products of Banach spaces

TL;DR

This paper demonstrates a straightforward method to represent elements in the completion of tensor products of Banach spaces as convergent series of elementary tensors.

Contribution

It provides a simple proof that all elements in the completion can be expressed as convergent series, simplifying understanding of tensor product structures.

Findings

Elements in the completion can be represented as convergent series of elementary tensors

The argument is notably simple and accessible

Applicable to any norm on the tensor product of vector spaces

Abstract

We show by a ridiculously simple argument that, for any norm on the tensor product of vector spaces, every element of the completion can be represented as a convergent series of elementary tensors.