14.7.2 Tensor Map Product Domain Tensor Space

Explore how tensor map products define operations within tensor spaces, establishing structure and transformations in algebraic contexts.

Tensor Map Product Domain Tensor Space is the single space V1 tensor V2 formed from the two domain factor spaces, serving as the actual carrier of the elements on which the tensor product of maps acts, distinct from the two factor spaces individually and equipped with its own universal property.

Formation of the Domain Tensor Space

Construction from the Domain Factor Spaces

Given domain factor spaces V1 and V2, the domain tensor space is formed as

V_{1} \otimes V_{2},

built through the ordinary tensor product construction, either as a quotient of the free vector space on the set V1 times V2 by the bilinear relations, or through any equivalent construction satisfying the same universal property.

Universal Property of the Domain Tensor Space

The domain tensor space is characterized by the property that every bilinear map out of V1 times V2 factors uniquely through the canonical map

\otimes : V_{1} \times V_{2} \to V_{1} \otimes V_{2},

a property invoked directly when constructing the tensor product of maps f tensor g from the bilinear map built out of f and g.

Elements of the Domain Tensor Space

Elementary Tensors

The domain tensor space is spanned by elementary tensors v tensor w with v from V1 and w from V2, and these elementary tensors are exactly the elements on which the elementary output rule of the tensor product of maps is directly specified.

General Elements and Their Non-Uniqueness of Representation

A general element of the domain tensor space is a finite sum of elementary tensors, and such a sum may typically be rewritten in several different ways as a sum of elementary tensors without changing the underlying element, a phenomenon the domain tensor space accommodates automatically through the bilinear relations built into its construction.

Coordinate Description of the Domain Tensor Space

Basis from the Factor Spaces

If e-1 through e-m is a basis of V1 and h-1 through h-n is a basis of V2, the elementary tensors e-i tensor h-j, ranging over all m n pairs of indices, form a basis of the domain tensor space, giving it dimension m n whenever V1 and V2 are finite-dimensional.

Coordinates of an Elementary Tensor

The coordinates of a general elementary tensor v tensor w with respect to this basis are the products of the coordinates of v with the coordinates of w, so that the coordinate vector of v tensor w is precisely the Kronecker product of the coordinate vector of v and the coordinate vector of w.

Role of the Domain Tensor Space in the Construction

Site of the Induced Map's Action

The domain tensor space is the actual space on which f tensor g acts as a linear map, distinct from V1 times V2, the set on which the underlying bilinear map beta is originally defined, since beta itself is not linear on V1 times V2 but only bilinear, while f tensor g is fully linear once transferred to the domain tensor space.

Relationship to the Codomain Tensor Space

The domain tensor space and the codomain tensor space of a tensor product of maps are generally distinct spaces, built from different pairs of factor spaces, except in the operator case where both the domain factor spaces and the codomain factor spaces coincide, causing the domain tensor space and codomain tensor space to coincide as well.