12.11.2 Tensor Restriction Target Subspace

Tensor Restriction Target Subspace is the smaller, deliberately chosen subspace of a tensor's original underlying vector space to which the tensor's multilinear action is confined during domain restriction, serving as the new effective vector space over which the restricted tensor is understood to operate.

Defining the Target Subspace

A Chosen Subspace of the Source Domain

Given a tensor $$A$$ built over a source domain $$V$$, the target subspace $$W$$ is any subspace satisfying $$W\subseteq V$$, selected specifically for the purpose of confining the tensor's arguments during restriction. The target subspace is chosen according to whatever context or application motivates the restriction, and different choices of $$W$$ lead to different restricted tensors.

The New Effective Vector Space

Once selected, the target subspace $$W$$ becomes the vector space over which the restricted tensor $$A_{|W}$$ is considered to be defined, replacing the original source domain $$V$$ as the relevant space for any further operations performed on the restricted tensor.

Requirements on the Target Subspace

Genuine Subspace Structure

For the restriction to be well defined, the target subspace $$W$$ must itself satisfy the axioms of a vector subspace within $$V$$, being closed under addition and scalar multiplication and containing the zero vector, since the restricted tensor must itself behave as a genuine tensor over $$W$$.

Compatible Dual Space

Corresponding to the target subspace $$W$$, an appropriately restricted dual space must also be identified for use in any covector slots of the tensor, ensuring that covariant and contravariant indices of the restricted tensor both draw on consistently related, smaller spaces.

Effect of the Target Subspace on the Restricted Tensor

Determining Which Components Survive

Choosing a basis of the target subspace $$W$$, possibly extended to a full basis of $$V$$, determines exactly which components of the original tensor $$A$$ are retained in the restricted tensor, namely those indexed entirely by directions lying within $$W$$.

Sensitivity to the Specific Subspace Chosen

Because different target subspaces generally correspond to different collections of basis directions, restricting the same original tensor $$A$$ to two different target subspaces typically produces two distinct restricted tensors, even though both originate from the identical starting tensor.

Practical Considerations in Selecting a Target Subspace

Guided by the Relevant Application

The target subspace is often chosen to reflect some meaningful constraint of the problem at hand, such as confining attention to directions tangent to a particular surface, or to a subset of coordinate directions considered relevant to a specific calculation.

Balancing Generality and Simplicity

Selecting a smaller target subspace typically simplifies the resulting restricted tensor, since fewer components survive the restriction, but doing so also discards information about the original tensor's behavior outside that subspace, so the choice of target subspace reflects a deliberate tradeoff between simplicity and completeness of information.

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