Universal Property, Maps and Isomorphisms of the Tensor Product

Abstract

This paper explores the tensor product of vector spaces through the lens of its universal property, drawing upon
fundamental concepts from Tensorial Calculus and Linear Algebra. The central goal is to demonstrate how the
tensor product serves as a tool to linearize bilinear maps , allowing any bilinear form to be translated into a unique
linear map over the newly generated vector space. The study revisits the construction of bases and dimensions for
tensor products, providing formal proofs for canonical isomorphisms between spaces of linear transformations and
tensors[cite: 9, 118]. Finally, practical applications are discussed regarding inner products in spaces such as Rn
and Mn(R), as well as the formalization of the complexification of real vector spaces and Lie algebras through
tensorial language.