1.2 Tensor Definition Foundations

Tensor Definition Foundations is the collection of interlinked concepts required to state a precise and general definition of a tensor, spanning the notions of vector spaces, dual spaces, multilinear maps, and the tensor product construction. It functions as the conceptual glossary from which the formal definition of a tensor is assembled, ensuring that each term used in that definition has itself already been given rigorous meaning.

Why a Tensor Cannot Be Defined in Isolation

A tensor is frequently introduced informally as "a quantity with multiple indices that transforms in a certain way," but this description presupposes an entire apparatus of prior concepts: what a vector space is, what a basis is, how coordinates change when the basis changes, what it means for a map to be linear, and what it means for a map to be linear in several variables simultaneously. Tensor Definition Foundations gathers precisely this apparatus, presenting it as a sequence of building blocks so that the eventual definition of a tensor rests on solid ground rather than on an appeal to intuition about arrays of numbers.

The Chain of Prerequisite Concepts

Vector Spaces and Their Basic Apparatus

The starting point is the vector space itself, together with the notion of a scalar field over which it is defined, a basis that allows every vector to be expressed uniquely as a linear combination of basis elements, and the associated coordinates that result from such an expression. Linear independence and linear combinations describe how vectors relate to one another within the space, while linear maps describe how one vector space relates to another while preserving this algebraic structure.

Dual Vector Spaces and Covectors

From any vector space, a second space can be constructed: its dual, consisting of all linear functionals that map vectors to scalars. Elements of the dual space are called covectors, and once a basis is chosen for the original space, a corresponding dual basis can be constructed for the dual space. The natural pairing between a vector and a covector — the scalar obtained by applying the covector to the vector — is the foundational operation that later distinguishes covariant from contravariant behavior under a change of basis.

Multilinear Maps

A multilinear map takes several vector arguments and is linear in each one separately, holding the others fixed. Bilinear maps take two arguments, trilinear maps take three, and the general notion extends to any number of arguments. Multilinear maps that produce a scalar are called multilinear forms, and these forms may additionally be classified as symmetric, if reordering the arguments leaves the result unchanged, or alternating, if swapping any two arguments reverses the sign of the result.

The expression above states the defining property of multilinearity in one argument: linearity holds independently in each slot of the map, with every other argument held fixed.

The Tensor Product Construction

The tensor product takes two or more vector spaces and produces a new vector space whose elements — elementary or decomposable tensors, and their linear combinations — encode multilinear relationships between the original spaces in a form that is itself linear. This construction is characterized by a universal property: any multilinear map out of the original spaces corresponds to a unique linear map out of their tensor product. This universal property is what makes the tensor product, rather than any particular multilinear map, the canonical setting in which tensors are defined.

Classification: Rank, Order, and Type

Once the tensor product construction is available, tensors can be classified by rank or order, referring to the total number of vector and covector factors involved, and by type, distinguishing how many of those factors are covariant (drawn from the dual space) versus contravariant (drawn from the original space). A tensor built from covariant factors alone, contravariant factors alone, or a mixture of both is described accordingly, and this classification determines how the tensor's components transform under a change of basis.

From Foundations to Formal Definition

With vector spaces, duality, multilinearity, and the tensor product all in place, a tensor can finally be defined with precision: as an element of a tensor product of copies of a vector space and its dual, or equivalently, as a multilinear map on the appropriate combination of vectors and covectors. Both formulations are equivalent, connected precisely through the universal property of the tensor product, and either may be adopted as the starting point depending on whether a coordinate-free or a map-based perspective is preferred. What matters is that, by the time this definition is reached, every term it contains has already been made precise by the foundational concepts assembled beforehand.