2.4.4 Tensor Vector Element Abstract Form

The Tensor Vector Element Abstract Form expresses vectors in multi-dimensional spaces using tensor algebra, linking linear algebra and advanced mathematics.

Tensor Vector Element Abstract Form is the basis-free description of a tensor as an intrinsic mathematical object, either as a multilinear map on vectors and covectors or as an element of a tensor product space characterized by a universal property, given without reference to any particular coordinate system, index labeling, or chosen basis. It is the complement to the coordinate form: where the coordinate form exhibits a tensor through an explicit array of numbers relative to a basis, the abstract form exhibits the same tensor as an object whose defining properties hold independently of any such choice.

The Multilinear Map Formulation

Definition

Let $V$ be a finite-dimensional vector space over a field $F$ . In abstract form, an element $T$ of the tensor space $T_{s}^{r} (V)$ is a function

T : \underset{r factors}{V^{*} \times \dots \times V^{*}} \times \underset{s factors}{V \times \dots \times V} \to F

satisfying linearity in each of its $r + s$ arguments separately, with all other arguments held fixed. No basis, index, or component array appears anywhere in this description; the object is characterized entirely by its behavior as a function.

What the Abstract Form Omits

The abstract form deliberately omits any statement of how $T$ would be written out numerically. Two tensors presented in abstract form are recognized as the same element if their defining functions agree on every input, with no mention of components required to state or verify this.

The Universal Property Formulation

Multilinear Maps Factoring Through the Tensor Product

An equivalent abstract characterization arises from the universal property of the tensor product: the space $T_{s}^{r} (V)$ may be identified with

\underset{r factors}{V \otimes \dots \otimes V} \otimes \underset{s factors}{V^{*} \otimes \dots \otimes V^{*}}

so that an element $T$ is, abstractly, a member of this tensor product space, defined up to the identifications forced by bilinearity of the tensor product symbol $\otimes$ itself.

Characterizing Property

This tensor product space is characterized, up to a unique isomorphism, by the property that every multilinear map out of the factors $V \times \dots \times V^{*}$ factors uniquely through it as a linear map. The abstract form of a tensor is therefore pinned down not by exhibiting its numerical values, but by the role it plays with respect to every such multilinear map.

Simple Tensors in Abstract Form

Elementary Products

A simple tensor is written abstractly as a tensor product of vectors and covectors,

T = v_{1} \otimes \dots \otimes v_{r} \otimes ω^{1} \otimes \dots \otimes ω^{s}

with no reference to which basis, if any, the individual factors $v_{i}$ and $ω^{j}$ are expressed in. The factors themselves may later be expanded in coordinates, but the abstract expression remains meaningful before any such expansion is performed.

General Elements

A general element in abstract form is a finite sum of such simple tensors, again without reference to coordinates:

T = \sum_{k = 1}^{N} v_{k} \otimes w_{k}

shown for type $(2, 0)$ , expressing the element as a superposition of elementary tensor products.

Operations in Abstract Form

Addition and Scalar Action Without Coordinates

Both tensor addition and scalar action are stated in abstract form purely in terms of the underlying functions or tensor-product elements:

(S + T) (ω^{1}, \dots, v_{s}) = S (ω^{1}, \dots, v_{s}) + T (ω^{1}, \dots, v_{s})

No index manipulation or component array is needed to define either operation; both are meaningful as soon as the abstract multilinear-map or tensor-product description is available.

Relationship to the Coordinate Form

Coordinate Form as a Consequence

Once a basis of $V$ is chosen, the abstract form yields the coordinate form by evaluation on basis vectors and dual basis covectors, or, equivalently, by expanding tensor product factors in that basis. The coordinate form is thus a derived description, always recoverable from the abstract form, while the abstract form carries no dependence on which coordinate form, if any, is subsequently extracted.

Why the Abstract Form Is Preferred for Theory

Because it involves no arbitrary choice of basis, the abstract form is the natural setting for stating and proving general facts about tensors, such as the vector space axioms, closure properties, and the behavior of tensor products, since such facts hold for the object itself rather than for one of its many possible coordinate representations. Coordinate computations remain indispensable for explicit numerical work, but they are properly understood as instances of the single, basis-independent object described by the abstract form.

Invariance as the Defining Feature

No Privileged Basis

The abstract form makes explicit that a tensor has no privileged representation: every basis of $V$ gives rise to a coordinate form, and all of these coordinate forms describe one and the same abstract element. The abstract form is precisely the invariant content shared by every one of these coordinate descriptions.