14.1.4 Tensor Map Product Evaluation Scope

Exploring how tensor map products are evaluated within their defined scope in algebraic structures.

Tensor Map Product Evaluation Scope is the specification of which elements of a tensor product a tensor product of linear maps is entitled to be evaluated on, together with the rules governing how that evaluation is carried out once an element is presented in a form other than a single elementary tensor.

Purpose of the Evaluation Scope

Evaluation on Elementary Tensors

The evaluation scope begins with the defining rule of the tensor product of maps: for elementary tensors, the map f tensor g acts by

(f \otimes g) (v \otimes w) = f (v) \otimes g (w) .

This rule fixes the evaluation scope on the generating set of the tensor product, and every other case of evaluation is reduced to repeated application of this rule.

Extension to General Elements by Linearity

Since a general element of a tensor product is a finite sum of elementary tensors, the evaluation scope extends by linearity: evaluating f tensor g on a sum means evaluating it term by term on each elementary tensor in that sum and adding the results,

(f \otimes g) (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} f (v_{i}) \otimes g (w_{i}) .

This extension is what places the full tensor product, rather than only its elementary tensors, inside the evaluation scope of the map.

Well-Definedness Across Representations

The Representation Problem

A single element of a tensor product typically admits many different expressions as a sum of elementary tensors. The evaluation scope is only meaningful if the value produced by the sum-by-sum rule above does not depend on which particular representation is used to compute it.

Guarantee from the Universal Property

Well-definedness of the evaluation scope across all representations is guaranteed by the universal property used to construct f tensor g in the first place: the map is defined as the unique factorization of a bilinear map through the tensor product, and a map obtained that way automatically assigns a single, representation-independent value to every element of the tensor product, including elements with no preferred decomposition into elementary tensors.

Boundaries of the Evaluation Scope

Elements Outside a Restricted Product

If the evaluation scope of f tensor g is deliberately restricted to a subspace of the full tensor product, for instance a subspace spanned by elementary tensors drawn from fixed subspaces of the two factors, then elements of the ambient tensor product lying outside that subspace fall outside the evaluation scope, and the map is not considered to act on them within that restricted context.

Partial Evaluation on One Factor

The evaluation scope also covers the partial application of only one of the two maps to an element of the tensor product, holding the other factor fixed. Tensoring f with the identity map restricts the evaluation scope to acting only through the first factor,

(f \otimes {id}_{W}) (v \otimes w) = f (v) \otimes w,

leaving the second factor evaluated by the identity and therefore unchanged.

Evaluation Scope Under Composition

Sequential Evaluation

When two tensor products of maps are composed, the evaluation scope of the composite agrees with evaluating the first tensor product of maps and then evaluating the second tensor product of maps on the result,

(f^{'} \otimes g^{'}) ((f \otimes g) (v \otimes w)) = f^{'} (f (v)) \otimes g^{'} (g (w)),

confirming that the evaluation scope of a composite tensor product of maps matches the evaluation scope obtained by combining the two individual maps directly, without needing to reconstruct the composite through its own universal property each time it is applied to an element.

Consistency with Rank and Image

The evaluation scope, restricted to elementary tensors of basis vectors, is exactly what determines the matrix entries of the induced map on the tensor product, and consequently determines its image and rank; no element outside the evaluation scope contributes to these invariants, since every element of the tensor product is reached by the linear extension described above.