14.7 Tensor Map Product Domain Structure

Tensor Map Product Domain Structure is the organization of everything specific to the source side of a tensor product of maps, encompassing the two domain factor spaces, the tensor product they form together, and the way elements of that tensor product serve as the raw inputs on which the induced map acts.

Constituents of the Domain Structure

The Domain Factor Spaces

At the base of the domain structure lie the two domain factor spaces V1 and V2, the individual source spaces of the maps f and g forming the factor map pair, each retaining its own identity as an ordinary vector space independent of any tensoring operation.

The Domain Tensor Product

Built from the domain factor spaces, the domain structure includes the tensor product

the single space on which the induced map f tensor g actually acts, formed from V1 and V2 according to the ordinary tensor product construction, complete with its universal property and its spanning set of elementary tensors.

Internal Organization of the Domain Structure

Elementary Tensors as the Generating Layer

The domain structure is organized around the elementary tensors v tensor w, with v from V1 and w from V2, since these generate the whole domain tensor product and are the elements on which the action of f tensor g is directly specified before being extended by linearity to the rest of the domain structure.

General Elements as the Complete Layer

Beyond elementary tensors, the domain structure includes every finite sum of elementary tensors, together with the equivalence of different sums that happen to represent the same element, a subtlety managed entirely by the universal property underlying the domain tensor product rather than by any separate bookkeeping device.

Domain Structure in Coordinates

Basis Assembly

Once bases are fixed for V1 and V2, the domain structure acquires a concrete coordinate description: a basis of the domain tensor product is assembled directly from the elementary tensors of basis vectors of V1 and V2, and every element of the domain structure is represented, relative to this basis, by a single coordinate vector indexed by pairs of basis indices.

Dimension of the Domain Structure

For finite-dimensional domain factor spaces of dimensions m and n, the domain structure has total dimension m n, matching the size of the square or rectangular matrix used to represent the induced tensor product map once a corresponding codomain structure is also fixed.

Domain Structure Under Composition and Restriction

Stability of the Domain Structure Under Composition

When composing two tensor products of maps, the domain structure of the composite is exactly the domain structure of the first tensor product of maps in the chain, since composition only modifies which map acts on the domain structure, not the domain structure itself.

Domain Structure Restricted to Subspaces

If U is a subspace of V1, the domain structure restricted to U tensor V2 forms a smaller domain structure in its own right, complete with its own elementary tensors and its own induced basis, nested inside the domain structure of the full tensor product V1 tensor V2, and this nesting is exactly what supports the analysis of invariant subspaces and restricted tensor products of maps throughout the broader theory.