9.3.1 Tensor Basis Element Set

Tensor Basis Element Set is the explicit, enumerable collection of individual simple (decomposable) tensor products — each one a single basis vector or dual basis covector taken from every tensor factor position — that together constitute the induced basis of a tensor space of a given type, together with the combinatorial rule for counting and indexing exactly how many such elements exist. It treats the induced tensor basis not as an abstract whole but as a concrete, listable set of individual basis elements, each one directly labeled by a specific combination of index values.

Enumerating the Element Set

One Element per Combination of Index Values

For a (p,q) tensor built from an n-dimensional space, the basis element set consists of every possible simple tensor eᵢ₁ ⊗ ⋯ ⊗ eᵢₚ ⊗ eʲ¹ ⊗ ⋯ ⊗ eʲq, one for each combination of values assigned independently to i₁, ..., iₚ and j₁, ..., jq, with each index ranging over 1 through n. The element set is therefore in exact one-to-one correspondence with the set of all such multi-index combinations.

Counting the Size of the Element Set

Because each of the p + q index slots independently ranges over n values, the total number of elements in the basis element set is n^{p+q}, matching the known dimension of the space of (p,q) tensors built from an n-dimensional space; this count is the direct combinatorial consequence of the element set's construction as all combinations of independently varying index labels.

Each Element Is a Single Simple Tensor

Distinguishing an Element From a General Tensor of the Type

Every member of the basis element set is a simple, or decomposable, tensor — a single tensor product of basis vectors and covectors with no summation involved — whereas an arbitrary tensor of that same type is, in general, a weighted sum over the entire element set and is typically not itself simple; the element set consists exclusively of these individual, unweighted building blocks, not of the more general tensors that can be built by combining them.

Coefficients Are Attached Externally, Not Part of the Set

The component array T^{i₁⋯}_{j₁⋯} that weights each basis element in an expansion is a separate piece of data from the element set itself; the basis element set is fixed once the underlying primal basis and its order are fixed, and it does not vary depending on which particular tensor is being expanded, only the coefficients attached to each of its fixed elements vary from one tensor to the next.

Indexing the Element Set for Practical Use

Multi-Indices as Labels for Individual Elements

Because listing every basis element by writing out its full tensor product notation becomes unwieldy for higher-order tensors, the element set is often indexed instead by a single combined multi-index, a tuple (i₁, ..., iₚ, j₁, ..., jq) standing in for the corresponding basis element; this indexing is purely a labeling convenience and does not change which elements belong to the set or how many there are.

Ordering the Element Set

Once an order is fixed on the underlying primal basis, a corresponding order can be imposed on the basis element set itself — commonly a lexicographic ordering of the multi-indices — which becomes relevant when the element set is used to arrange tensor components into a single flattened list or array, such as when a rank-2 tensor's basis element set is ordered to match the row-and-column layout of its matrix representation.

Diagram of the Basis Element Set for a Rank-2 Tensor

Why the Element Set Is Treated as a Single Fixed Object

Same Element Set Serves Every Tensor of the Given Type

Because the element set depends only on the underlying primal basis and the tensor type (p,q), not on any particular tensor being expressed, the same fixed element set serves as the expansion basis for every tensor of that type simultaneously; this is what allows a single index notation convention, once the underlying basis is fixed, to describe an arbitrary tensor of a given type without re-deriving the element set anew for each one.

The Element Set's Size Governs Notational and Computational Cost

The count n^{p+q} of the basis element set directly determines how many independent numbers are needed to specify an arbitrary tensor of that type in the given basis, and correspondingly how many terms appear when a general such tensor is expanded, tying the combinatorics of the basis element set directly to both the storage cost and the computational cost of working with tensors of high order or built from high-dimensional spaces.