14.22.2 Tensor Map Product Factor Notation

Tensor Map Product Factor Notation simplifies tensor mappings using factorized notation, streamlining algebraic operations in tensor algebra.

Tensor Map Product Factor Notation is the convention governing how the individual maps being tensored together are named and indexed, matching each map's label to the label of the vector space it acts on so that an expression such as $f_{1} \otimes \dots \otimes f_{n}$ remains unambiguous even as the number of tensored factors grows beyond two.

Naming the Two Factors in the Basic Case

Letter Pairs by Context

For a tensor product of exactly two maps, common letter pairings include $f, g$ in a general algebraic setting, $T, S$ when both maps are operators on their respective spaces, and $φ, ψ$ when both maps are functionals; none of these pairings carries mathematical content beyond readability, and any two distinct symbols suffice to name the two factors.

Matching Domain and Codomain Labels

Whichever letters are chosen for the maps, the corresponding domain and codomain spaces are conventionally labeled with related symbols, for instance $f : V \to V^{'}$ paired with $g : W \to W^{'}$ , so that the correspondence between a map's letter and its associated space's letter, here $f$ with $V$ and $g$ with $W$ , is visible without needing to consult a separate legend.

Extending to More Than Two Factors

Indexed Letters for a Family of Factors

When tensoring $n$ maps together, a single letter with a numerical subscript, $f_{1}, f_{2}, \dots, f_{n}$ , replaces the practice of choosing $n$ unrelated letters, since distinct letters become impractical to generate and to remember once $n$ exceeds two or three.

Matching Indexed Spaces

The corresponding domain and codomain spaces are indexed the same way, $V_{1}, \dots, V_{n}$ and $V_{1}^{'}, \dots, V_{n}^{'}$ , with $f_{k} : V_{k} \to V_{k}^{'}$ for each $k$ , giving the combined map

f_{1} \otimes \dots \otimes f_{n} : V_{1} \otimes \dots \otimes V_{n} \to V_{1}^{'} \otimes \dots \otimes V_{n}^{'}

with every index consistently threading through the map name, the domain space, and the codomain space simultaneously.

Compact Notation for the Whole Family

Big-Tensor Notation

Analogous to the summation sign, a big-tensor symbol is sometimes used to abbreviate a long tensor product of many indexed factors,

⨂_{k = 1}^{n} f_{k}

standing for $f_{1} \otimes \dots \otimes f_{n}$ exactly as the summation sign stands for a long sum, and used most often when $n$ is left as a general or unspecified natural number rather than a small fixed integer.

Notation When All Factors Coincide

If every factor is the same map $f$ , the big-tensor notation collapses to the tensor power notation $f^{\otimes n}$ , and the corresponding indexed spaces collapse to $V^{\otimes n}$ , removing the need for indices altogether once all factors are identical.

Disambiguating Factor Notation From Other Uses of Subscripts

Not to Be Confused With Component Indices

The subscripted factor labels $f_{1}, \dots, f_{n}$ , naming which map occupies which position in the tensor product, are a distinct use of subscripts from the lower indices $i, j$ used in component notation to label basis vectors; a careful presentation keeps these two uses of subscript notation visually or contextually separated, for instance by reserving $k, l$ for factor labels and $i, j, a, b$ for basis indices within each factor.

Not to Be Confused With Operator Position Subscripts

The factor notation described here, naming the maps $f_{k}$ being tensored, is also distinct from the operator notation convention in which a subscript such as $T_{1}$ denotes an operator embedded into a fixed combined space acting only on one factor; the former subscript names a map among several being combined, while the latter subscript names a position within an already-formed combined space, and the two uses should not be conflated even though both employ small numerical subscripts.