12.11.1 Tensor Restriction Source Domain

Tensor Restriction Source Domain is the original, larger vector space over which a tensor was initially defined before any domain restriction was applied, serving as the reference point from which a smaller subspace is selected and against which the validity of the restriction operation is judged.

Identifying the Source Domain

The Original Vector Space Before Restriction

For a tensor $$A$$ of type $$(p,q)$$, the source domain is the vector space $$V$$, together with its dual $$V^{*}$$, over which $$A$$ is originally defined, prior to selecting any subspace $$W$$ for the purposes of restriction. It is the domain from which the restriction operation draws its starting point.

Necessity of a Well-Defined Source

Domain restriction is only meaningful relative to a clearly identified source domain, since the subspace $$W$$ to which a tensor is restricted must be understood as a subspace specifically of this particular source domain $$V$$, rather than an arbitrary vector space unrelated to where $$A$$ was originally defined.

Role of the Source Domain in the Restriction Operation

Anchoring the Subspace Selection

Every choice of subspace $$W$$ used in a domain restriction is made relative to the source domain $$V$$, requiring the containment relationship $$W\subseteq V$$ to hold. Without a well-specified source domain, this containment relationship, and therefore the entire restriction operation, could not be properly formulated.

Preserving the Original Multilinear Rule

The source domain is also where the original tensor's defining multilinear formula lives; restriction does not alter this formula but merely limits which inputs from the source domain's subspace are supplied to it, so the source domain remains the ultimate origin of the rule governing how the restricted tensor computes its outputs.

Distinguishing the Source Domain from the Target Subspace

Two Distinct Roles

The source domain $$V$$ and the subspace $$W$$ play distinct roles in the restriction operation: the source domain is the larger space being narrowed, while the subspace is the smaller space to which the tensor's arguments become confined. Conflating the two would misrepresent the direction and meaning of the restriction.

The Source Domain Remains Fixed Across Different Restrictions

A single tensor may be restricted to several different subspaces for different purposes, each restriction sharing the same fixed source domain $$V$$ as its starting point, even though the resulting restricted tensors differ according to which subspace was selected.

Practical Significance

Establishing Context Before Restriction

Before applying domain restriction, it is necessary to establish clearly what the source domain of the tensor in question actually is, since this identification determines which subspaces are even eligible to serve as valid targets for restriction.

Ensuring Meaningful Comparisons Between Restrictions

Because different restrictions of the same tensor share the same source domain, comparisons between such restrictions, for instance examining how the tensor behaves on one subspace versus another, remain meaningful precisely because both restrictions trace back to the same underlying original tensor and source domain.

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