Five basic lemmas for symmetric tensor products of normed spaces

TL;DR

This paper extends five fundamental lemmas to symmetric tensor products of normed spaces, providing essential tools for the metric theory of symmetric tensor products and polynomial ideals.

Contribution

It introduces symmetric versions of key lemmas in tensor product theory, enhancing the mathematical framework for symmetric tensor spaces.

Findings

Symmetric approximation lemma established

Extension and embedding lemmas adapted for symmetric tensor products

Applications to metric theory and polynomial ideals demonstrated

Abstract

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 lemmas. Some application of these tools to the metric theory of symmetric tensor products and to the theory of polynomials ideals are given.