14.5.3 Tensor Linear Functional Product Covector Relation

Explore how tensor products connect linear functionals and covectors in algebraic structures through their functional relations.

Tensor Linear Functional Product Covector Relation is the correspondence identifying a tensor product of linear functionals with a single covector on the tensor product space, meaning an element of the dual space of the tensor product rather than merely a pairing of two separate functionals.

The Basic Identification

Functionals as Covectors

Each individual linear functional phi on V is itself a covector on V, an element of the dual space V star, and the covector relation extends this identification to tensor products: the elementary tensor phi tensor psi, though built from two covectors on two different spaces, is identified with a single covector on the tensor product space V tensor W, via

φ \otimes ψ \in {(V \otimes W)}^{*} .

The Canonical Map Realizing the Relation

The covector relation is realized by the canonical linear map

V^{*} \otimes W^{*} \to {(V \otimes W)}^{*},

sending phi tensor psi to the covector acting on elementary tensors by phi(v) times psi(w), an injective map in general and an isomorphism whenever V and W are finite-dimensional.

Consequences of the Covector Relation

Every Elementary Product Is a Covector, Not Every Covector Is a Product

The covector relation shows that every elementary tensor of functionals is a covector on the tensor product, but in the finite-dimensional case the relation is surjective as well, so every covector on V tensor W arises this way, though typically not as a single elementary product but only as a finite sum of such elementary products, corresponding to the fact that a general covector has rank possibly greater than one when regarded as a bilinear form.

The Covector Relation and Rank

The rank of a covector on V tensor W, viewed through the covector relation as a bilinear form on V times W, measures the minimal number of elementary tensor products of functionals needed to express it, and a covector of rank one is exactly the image, under the covector relation, of a single elementary tensor phi tensor psi.

Compatibility with Basis Structures

Dual Basis Covectors

If e-1 through e-m is a basis of V with dual basis e-1 star through e-m star, and h-1 through h-n is a basis of W with dual basis h-1 star through h-n star, the covector relation sends each elementary tensor e-i star tensor h-j star to the covector on V tensor W dual to the basis elementary tensor e-i tensor h-j, so the covector relation carries the induced dual basis of the tensor product of duals precisely onto the dual basis of the tensor product space.

Coordinate Expression of the Covector Relation

Under this correspondence, a general element of V star tensor W star, expressed in coordinates as a matrix of coefficients with respect to the dual bases, corresponds through the covector relation to the covector on V tensor W whose action on a basis elementary tensor e-i tensor h-j is read directly from the (i, j) entry of that coefficient matrix.

Interaction with Composition and Pullback

Pullback of Covectors

If f is an operator on V and g is an operator on W, the covector relation is compatible with pullback in the sense that the covector corresponding to (phi composed with f) tensor (psi composed with g) equals the pullback, along f tensor g, of the covector corresponding to phi tensor psi, so pullback of covectors along operator products matches exactly the pullback of the underlying functionals along the individual operators.

Relation to the Double Dual

In the finite-dimensional case, composing the covector relation with the canonical identification of V with its double dual shows that elements of V tensor W can themselves be regarded as covectors on V star tensor W star, and under this further identification, the covector relation for functionals on V and W corresponds symmetrically to the analogous relation for vectors in V and W acting as covectors on the dual tensor product, exhibiting the covector relation as a special case of this broader duality between a space and the dual of its dual.