Tensor spaces - the basics

TL;DR

This paper introduces fundamental concepts of tensor products of vector spaces, focusing on linear algebraic and combinatorial methods relevant for applied research in multilinear algebra and tensor analysis.

Contribution

It provides a comprehensive overview of tensor product theory, including inner products and direct sums, tailored for researchers in applied mathematics and related fields.

Findings

Clarifies tensor product constructions and properties

Explains inner products on tensor spaces

Provides foundational knowledge for applied tensor analysis

Abstract

We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3) Tensor products of vector spaces; (4) Tensor products of matries; (5) Inner products on tensor spaces; (6) Direct sums and tensor products; (7) Background concepts and notation.