2.19 Tensor Abstract Vector Space Context

Tensor Abstract Vector Space Context is the treatment of tensor algebra built entirely from the axioms of a vector space over an arbitrary field, without invoking any structure specific to a particular field such as the real numbers or the complex numbers. It is the coordinate-free, basis-independent setting in which a tensor is defined by its universal property, as an object representing multilinear maps, rather than by an array of numerical components subject to a transformation law. This context supplies the common algebraic skeleton, applicable uniformly to real vector spaces, complex vector spaces, or vector spaces over any other field, from which the more specialized contexts, such as the real or complex vector space contexts, are obtained by adding field-specific structure on top.

The Abstract Vector Space as Starting Point

Field-Independent Axioms

An abstract vector space V over a field F is a set equipped with vector addition and scalar multiplication by elements of F, satisfying the usual axioms of associativity, commutativity of addition, distributivity, and the existence of an additive identity and additive inverses. Nothing in these axioms refers to real numbers, complex numbers, continuity, or any topological or metric notion; F may be any field, including finite fields, and the vector space axioms remain exactly the same.

No Privileged Basis

In the abstract context, V is not presented with a preferred coordinate system. A basis may be chosen for computational convenience, but the vector space itself, and any tensor built from it, is defined independently of that choice; a change of basis is regarded as a relabeling of the same abstract object, not a change in the object itself.

Tensors Defined by Universal Property

The Universal Property of the Tensor Product

Given abstract vector spaces V and W over F, their tensor product V ⊗ W is defined, up to unique isomorphism, as the vector space equipped with a bilinear map ⊗ : V × W → V ⊗ W such that every bilinear map B : V × W → U into any vector space U factors uniquely as B = L ∘ ⊗ for a linear map L : V ⊗ W → U.

This property characterizes the tensor product entirely by how it interacts with other spaces and maps, with no reference to components, matching the abstract, coordinate-free character of the surrounding context.

Tensors as Elements, Not Arrays

Within this context, a (p, q)-tensor is an element of V^{⊗p} ⊗ (V^{*})^{⊗q}, the abstract tensor product space, defined without ever writing down a component array. Component representations are recovered only after a basis is introduced, at which point the tensor's coordinates in that basis are computed by pairing the abstract element against the basis and dual basis vectors.

Consequences of Working Abstractly

Basis Independence of Definitions and Theorems

Because every construction in this context, the tensor product, contraction, symmetrization, and the wedge product, is defined through universal properties or through maps that are manifestly independent of a choice of basis, any theorem proved in the abstract vector space context automatically holds in every basis and specializes correctly once a particular field or basis is fixed.

Separation of Algebraic and Field-Specific Structure

The abstract context isolates what is purely algebraic about tensors, such as multilinearity, symmetry, and contraction, from what depends on extra structure carried by a specific field, such as complex conjugation on C, an ordering on R, or a norm derived from an inner product. This separation is what allows the complex vector space context, and any other field-specific context, to be presented as a targeted extension of the abstract theory rather than as an independent redevelopment of it.

Relation to Concrete Component Computation

Choosing a Basis to Compute

When explicit computation is required, a basis e_1, ..., e_n of V is selected, and the abstract tensor is expressed as a linear combination of basis tensor products with coefficients called its components. The abstract context guarantees in advance that any two such expressions, obtained from two different bases, are related by the standard tensor transformation law, so the choice of basis never affects the underlying tensor, only its numerical description.

Dimension as the Only Numerical Invariant Needed

For a finite-dimensional abstract vector space V of dimension n, the dimension of the tensor space V^{⊗p} ⊗ (V^{*})^{⊗q} is n^{p+q}, a fact derivable purely from counting basis tensor products and requiring no further assumption about the field F beyond it supporting a finite basis.

Diagrammatic Summary

The diagram shows the abstract vector space context as the common root from which field-specific contexts, such as the real and complex vector space contexts, extend by adding structure particular to that field while retaining every basis-independent construction defined at the abstract level.