4.23.5 Tensor Multilinear Evaluation Notation

Tensor Multilinear Evaluation Notation is the set of conventions for writing down the act of feeding specific vectors into a multilinear map and reading off its output, spanning direct functional application, pairing-bracket notation borrowed from dual spaces, and the expanded component-sum notation used when a computation is to be carried out explicitly in coordinates.

Direct Functional Evaluation

The Basic Form

The most immediate evaluation notation simply applies the function symbol to a parenthesized list of specific vectors, f(v₁,...,vₙ), denoting the single element of W obtained by evaluating f at that particular tuple; this is the notation used whenever the multilinear map is being treated as an operation actually being carried out, as opposed to being discussed abstractly as an object in its own right.

Evaluation on Basis Vectors

A frequently occurring special case evaluates f on a tuple drawn entirely from chosen bases, f(e_{i₁},...,e_{iₙ}), which by definition equals the component array entry T_{i₁...iₙ}; this notational identification, f(e_{i₁},...,e_{iₙ}) = T_{i₁...iₙ}, is the bridge connecting the functional-evaluation notation to the component-array representation.

Pairing-Bracket Notation

Duality Notation for Scalar-Valued Evaluation

When a multilinear map is scalar-valued and is being regarded as an element of a dual tensor product, evaluation is often written using pairing brackets borrowed from the notation for dual spaces, ⟨f, v₁ ⊗ ... ⊗ vₙ⟩, denoting the result of applying the linear functional corresponding to f to the elementary tensor v₁ ⊗ ... ⊗ vₙ; this is notationally distinct from, though numerically equal to, f(v₁,...,vₙ), since the pairing bracket emphasizes evaluation of a functional on a tensor-product element rather than direct application of a multilinear function to a tuple.

Consistency With the Universal Property

The equality ⟨f, v₁ ⊗ ... ⊗ vₙ⟩ = f(v₁,...,vₙ) is precisely the commutative relation guaranteed by the tensor product's universal property, restated using pairing notation; writing evaluation this way makes explicit that the value does not depend on which of the two equivalent perspectives, multilinear map or dual tensor functional, is used to compute it.

Component-Sum Evaluation Notation

Expanding in Coordinates

When vectors are given in coordinates, vⱼ = ∑ᵢ vⱼⁱ e_i, evaluation is expanded into an explicit sum over all combinations of basis indices,

This is the notation used whenever an evaluation is to be actually carried out numerically, converting the abstract application f(v₁,...,vₙ) into an explicit finite computation involving only ordinary numbers.

Einstein Summation Shorthand for Evaluation

Using the summation convention, the same evaluation is written without explicit summation symbols, f(v₁,...,vₙ) = v₁^{i₁}⋯vₙ^{iₙ} T_{i₁...iₙ}, with repeated indices summed implicitly; this compressed form is preferred whenever evaluation formulas are written repeatedly, since it removes the visual overhead of the summation symbols while retaining full computational content.

Evaluation of the Induced Linear Map

Distinguishing f from f̃

When a multilinear map f and its induced linear map f̃ on the tensor product are both in play, evaluation notation must distinguish f(v₁,...,vₙ), direct multilinear evaluation on a tuple, from f̃(t), linear evaluation of the induced map on a general tensor t, which need not be an elementary tensor; the relation f̃(v₁ ⊗ ... ⊗ vₙ) = f(v₁,...,vₙ) connects the two notations exactly on elementary tensors, while f̃ alone carries meaning on the more general elements of the tensor product where f itself is not directly defined.

Evaluation on a General Tensor

For t = ∑ₖ v_{1,k} ⊗ ... ⊗ v_{n,k} a general (non-elementary) tensor, evaluation notation for f̃(t) expands using linearity of f̃, f̃(t) = ∑ₖ f(v_{1,k},...,v_{n,k}), reducing evaluation on a general tensor to a sum of ordinary multilinear evaluations on the elementary tensors appearing in some chosen representation of t.

Evaluation Notation for Partially Applied Maps

Evaluating a Curried Map

When f has been curried by fixing some arguments, evaluation notation for the resulting partially applied map follows the same functional form applied to the remaining arguments, f(v, -)(w) = f(v, w), making explicit that applying the curried map to the remaining argument reproduces the original full evaluation.

Chained Evaluation Notation

For a fully curried multilinear map viewed as an iterated linear map, f ∈ Hom(V₁, Hom(V₂, ..., Hom(Vₙ, W))), evaluation is written as a chain of applications, f(v₁)(v₂)⋯(vₙ), with each application peeling off one Hom-layer at a time; this notation is used specifically when the iterated linear structure of f, rather than its flat multilinear structure, is the object of discussion.

Choosing Evaluation Notation for the Task

Abstract Reasoning Favors Pairing Brackets

Discussions centered on the universal property, on tensor rank, or on the relationship between multilinear maps and elements of a dual tensor product are typically clearest using pairing-bracket notation, since it keeps the tensor-algebraic perspective visually explicit throughout the argument.

Explicit Computation Favors Component Sums

Direct numerical work, verifying a specific identity, computing a value for a particular application, is clearest using the fully expanded component-sum notation or its Einstein-summation shorthand, since these forms map most directly onto the arithmetic actually being performed.