6.3.2 Tensor Order Argument Count

Tensor Order Argument Count refers to the number of indices required to uniquely identify a tensor component in a given space.

Tensor Order Argument Count is the same integer k as the tensor order slot count, arrived at instead from the multilinear-map viewpoint: a tensor of order k is a function T : V₁ × V₂ × ... × V_k → F accepting exactly k vector-space arguments and returning a scalar, and the argument count is simply the arity of this function, the number of separate inputs it must be given before it produces a scalar output. Where the slot count is read off index notation by counting index symbols, the argument count is read off the functional definition by counting how many separate vector-space inputs the map requires, and the two counting methods always agree for the same tensor.

The Multilinear-Map Definition

A Tensor as a Function of Several Vector Arguments

A type (p, q) tensor, viewed abstractly, is a function taking p covector arguments and q vector arguments (or the dual convention, depending on the source) and returning a scalar:

T : \underset{p}{\times} V^{*} \times \underset{q}{\times} V \to F

The argument count is the total number of factors in this domain, p + q, exactly matching the slot count computed from index notation.

Multilinearity as the Defining Requirement

Within this view, each of the k arguments must be filled linearly: holding all but one argument fixed, T must depend linearly on the remaining one; the argument count k is the number of positions at which this linearity requirement is separately imposed.

T (v, \dots, a u_{1} + b u_{2}, \dots) = a T (v, \dots, u_{1}, \dots) + b T (v, \dots, u_{2}, \dots)

illustrated at a single argument position, with the same requirement applying independently at every one of the k positions.

Comparing the Two Counting Perspectives

Index Notation Versus Functional Notation

Index notation records the argument count implicitly, as the number of free index symbols attached to T; functional notation records it explicitly, as the number of blank argument slots in T( _, _, ..., _ ). Converting between the two is direct: each index symbol corresponds to exactly one functional argument, in the same order.

Side-by-Side Diagram

Why Two Perspectives Are Kept Distinct

Although the two counting methods always produce the same number k, they emphasize different aspects of a tensor: index notation is convenient for explicit component computations and transformation formulas, while functional notation is convenient for abstract, coordinate-free arguments about multilinearity, universal properties, and currying.

Currying and Partial Application

Reducing Argument Count by Partial Application

Within the functional view, supplying a value for one of the k arguments while leaving the rest open produces a new multilinear map of argument count k - 1; this operation, called currying or partial application, has no direct analogue in the pure index-counting view beyond noting that one index has been replaced by a fixed value rather than left free.

T (v_{0}, •, \dots, •) : \underset{k - 1}{\times} V \to F

Argument Count Under Composition With the Universal Property

The universal property of the tensor product states that every multilinear map of argument count k factors uniquely through a linear map from V₁ ⊗ ... ⊗ V_k, a fact stated most naturally using argument-count language, since it concerns the arity of the map being universally represented rather than the specific index symbols used to write it.

Argument Count Under Operations

Argument Count of a Tensor Product

argcount (A \otimes B) = argcount (A) + argcount (B)

matches the slot-count addition rule exactly, since forming a tensor product of two multilinear maps produces a new map whose domain is the product of the two original domains.

Argument Count After Contraction

Contracting one upper argument against one lower argument, in functional terms, means composing T with a canonical pairing map on those two argument positions and summing, which reduces the argument count by exactly 2, mirroring the slot-count reduction rule under contraction.

Why the Argument-Count Perspective Is Useful

Supporting Coordinate-Free Reasoning

Because the argument-count view makes no reference to a chosen basis or index labels, it is the preferred perspective for proofs and constructions — such as the universal property of the tensor product — that are meant to hold independently of any coordinate system.

Clarifying the Meaning of Partial Contraction and Evaluation

Describing a tensor as accepting k arguments makes explicit that evaluating it on fewer than k arguments produces not a number but another tensor of reduced argument count, a fact that is easy to state precisely in functional language and only indirectly visible in pure index notation.