9.1.3 Tensor Basis Expansion Scope

Tensor Basis Expansion Scope is the extent to which the formula expressing a tensor as a sum of basis tensor products, weighted by its components, actually reconstructs every tensor of the relevant type — full scope when the chosen vectors and covectors genuinely form a complete basis of the space and its dual, and reduced scope, covering only a subspace of possible tensors, whenever the chosen expansion set is incomplete or the underlying space itself is available only locally, as within a single coordinate chart on a manifold. It asks, specifically, how far the reconstruction formula T = T^{i₁⋯} e_{i₁} ⊗ ⋯ ⊗ e^{j₁} ⊗ ⋯ can be trusted to recover the tensor it claims to expand, rather than how far the basis vectors themselves are individually defined.

Full Expansion Scope From a Genuine Basis

Completeness Guarantees Every Tensor Is Reachable

When {eᵢ} is an actual basis of V — linearly independent and spanning — the induced family {eᵢ₁ ⊗ ⋯ ⊗ eᵢₚ ⊗ eʲ¹ ⊗ ⋯ ⊗ eʲq} is a basis of the space of (p,q) tensors, and the expansion formula reconstructs every such tensor exactly, with no tensor of that type lying outside what the expansion can represent:

The expansion scope here is the entire space of (p,q) tensors built from V, matching the unrestricted algebraic scope of the underlying basis of V itself.

Full Scope Requires Both Factors of the Basis to Be Complete

Because tensor components require both the primal basis {eᵢ} and its dual {eⁱ}, full expansion scope is only guaranteed when both are genuine, complete bases in the sense described; supplying a complete basis of V while using an incomplete or ad hoc collection of covectors in place of the true dual basis narrows the expansion scope even though the vector-side basis is entirely adequate.

Reduced Scope From an Incomplete Expansion Set

Spanning Only a Subspace

If the vectors chosen for expansion span only a proper subspace W ⊂ V rather than the whole of V, the induced tensor products span only the corresponding subspace of the full tensor space, and any tensor with a component outside that subspace cannot be reconstructed by the expansion at all; the expansion scope in this case is strictly smaller than the full space of tensors, equal only to those built entirely from W and its relevant dual directions.

Partial Bases Encountered in Practice

An incomplete expansion set arises naturally whenever a calculation restricts attention to a distinguished subspace — such as the tangent directions along a submanifold, or the eigenspace of a particular linear operator — and expands only tensors that are known in advance to lie within that subspace; recognizing that the expansion scope has been deliberately narrowed in such cases is what prevents an accidental attempt to represent a tensor with components genuinely outside the chosen subspace using an expansion that cannot reach them.

Scope Limited by the Chart When Expanding Tensor Fields

The Coordinate Basis Restricts Expansion to Its Own Domain

When the basis used for expansion is a coordinate basis {∂/∂xⁱ} derived from a chart, the expansion of a tensor field in that basis is valid only at points within the chart's own domain, since the coordinate basis vectors themselves are undefined outside it; the expansion scope for a tensor field in a given chart is therefore the intersection of the chart's geometric scope with the region where the tensor field itself is defined.

Different Charts Yield Different, Equally Valid Expansions

Within the overlap of two charts, the same tensor field admits two different expansions, one in each chart's coordinate basis, related by the coordinate change transformation; neither expansion has scope beyond its own chart's domain, and reconstructing the tensor field's expansion over a region spanning both charts requires assembling the two chart-local expansions rather than extending either one past its own scope.

Diagram of Full Versus Reduced Expansion Scope

Practical Implications of Tracking Expansion Scope

Verifying Sufficiency Before Trusting an Expansion

Before relying on a claimed expansion to represent an arbitrary tensor of a given type, confirming that the expansion set used is actually complete (or, if not, explicitly identifying the smaller subspace it does cover) prevents the error of assuming a tensor has been fully captured when in fact only a restricted piece of it lies within the expansion's actual scope.

Combining Local Expansions Into a Global Description

Just as a single chart's coordinate basis expansion covers a tensor field only within that chart's domain, assembling a complete description of a tensor field over an entire manifold from local coordinate expansions requires the same overlap-consistency check used for basis and coordinate scope generally, confirming that the locally expanded pieces, converted appropriately across chart transitions, agree wherever their individual expansion scopes overlap.