Algebraic Tensor Products Revisited: Axiomatic Approach

TL;DR

This paper reviews algebraic tensor products of linear spaces through an axiomatic approach, emphasizing the universal property and its various isomorphic interpretations to clarify their foundational aspects.

Contribution

It provides an expository, axiomatic perspective on tensor products, unifying different construction methods via the universal property.

Findings

Clarifies the universal property as a unifying concept

Reduces concrete constructions to isomorphic interpretations

Enhances understanding of tensor product foundations

Abstract

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz., the universal property, which is based on a pair of axioms.