14.8.2 Tensor Map Product Codomain Tensor Space

Tensor Map Product Codomain Tensor Space is the single space W1 tensor W2 formed from the two codomain factor spaces, serving as the actual carrier of every output produced by a tensor product of maps, distinct from the two factor spaces individually and equipped with its own universal property.

Formation of the Codomain Tensor Space

Construction from the Codomain Factor Spaces

Given codomain factor spaces W1 and W2, the codomain tensor space is formed as

built through the ordinary tensor product construction, exactly as with the domain tensor space, but drawing on W1 and W2 in place of the domain factor spaces.

Universal Property Available to the Codomain Tensor Space

Because the codomain tensor space is itself a tensor product, it too satisfies the universal property that every bilinear map out of W1 times W2 factors uniquely through the canonical map into W1 tensor W2, a property the codomain tensor space may itself supply if it later serves as the domain tensor space for some further tensor product of maps in a composition chain.

Elements of the Codomain Tensor Space

Elementary Tensors as the Generating Set

The codomain tensor space is spanned by elementary tensors w1 tensor w2 with w1 from W1 and w2 from W2, and among these, the elementary tensors of the specific form f(v) tensor g(w) are exactly the ones directly produced by the elementary output rule of the tensor product of maps.

General Elements Beyond the Image

The codomain tensor space contains, in general, elements that are not outputs of f tensor g at all, since the image of f tensor g may be a proper subspace of the full codomain tensor space; the codomain tensor space itself, however, is simply the entire tensor product W1 tensor W2, independent of which particular subspace happens to be reached by any specific tensor product of maps.

Coordinate Description of the Codomain Tensor Space

Basis from the Factor Spaces

If h-1 through h-p is a basis of W1 and k-1 through k-q is a basis of W2, the elementary tensors h-i tensor k-j, ranging over all p q pairs of indices, form a basis of the codomain tensor space, giving it dimension p q whenever W1 and W2 are finite-dimensional.

Coordinates of an Output

The coordinates of an output f(v) tensor g(w), with respect to this basis, are the products of the coordinates of f(v) with the coordinates of g(w), so the coordinate vector of any elementary tensor output is precisely the Kronecker product of the coordinate vector of f(v) and the coordinate vector of g(w).

Role of the Codomain Tensor Space in the Construction

Site Where Outputs Are Assembled

The codomain tensor space is the space in which the values f(v) tensor g(w) are formed using ordinary tensor multiplication within W1 tensor W2, a role that requires no universal property to be invoked, in contrast to the domain tensor space, whose universal property is what makes the entire construction of f tensor g possible in the first place.

Relationship to the Domain Tensor Space in Composition

When two tensor products of maps are composed, the codomain tensor space of the first becomes the domain tensor space of the second, and this transition requires the codomain tensor space, initially serving only as a passive receiving space, to also satisfy the active universal property demanded of a domain tensor space, a role it is always capable of fulfilling since every tensor product space possesses this universal property regardless of whether it happens to be used as a domain or a codomain in any particular instance.