4.5.3 Tensor Argument Slot Input Element

A tensor argument slot input element assigns values to slots, enabling precise mathematical operations in tensor algebra.

Tensor Argument Slot Input Element is the specific vector or covector that is actually substituted into a given argument slot when a tensor's multilinear map is evaluated on a particular tuple of inputs. Where the slot itself is an abstract position bound to a vector space, the input element is the concrete member of that space chosen to occupy the slot at the moment of evaluation, and it is the object on which the slot's linearity condition is directly exercised.

Formal Definition

The Element Filling a Slot

For a tensor represented as a multilinear map

T : V_{1} \times \dots \times V_{k} \to F

evaluated on a tuple $(v_1, \ldots, v_k)$, each $v_i \in V_i$ is the input element occupying slot $i$. The input element is required to satisfy exactly one condition: membership in the vector space bound to that slot. No other constraint (such as norm, positivity, or basis expansion) is imposed by the tensor structure itself; any element of the correct space is an admissible input element.

Input Elements Versus Slot Definitions

A slot is a fixed, unchanging part of the tensor's structure; an input element is a variable, freely chosen quantity supplied at evaluation time. The same slot can receive infinitely many different input elements across different evaluations of $T$, while the slot's variance, position, and bound vector space remain constant throughout.

Behavior of Input Elements Under Linearity

Additive and Scalar Combinations

Because the map is linear in each slot, an input element that is itself expressed as a linear combination of other elements distributes across the evaluation:

T (\dots, a u + b w, \dots) = a T (\dots, u, \dots) + b T (\dots, w, \dots)

This is what allows any input element to be expanded in a chosen basis of its slot's vector space, and for the evaluation of $T$ to be reduced to a finite sum over the tensor's components, weighted by the coordinates of each input element.

Basis Coordinates as Canonical Input Elements

The basis vectors of a slot's vector space serve as the canonical set of input elements: substituting a full set of basis vectors, one into each slot, produces exactly the components of the tensor. Every other admissible input element is, from the perspective of the multilinear structure, nothing more than a specific linear combination of these canonical elements.

Constraints and Consistency

Type Matching

An input element intended for a covariant slot must belong to $V$, and an input element intended for a contravariant slot must belong to $V^{*}$. This type matching is enforced by the underlying vector space bound to the slot, and any mismatch renders the evaluation undefined rather than merely producing an unexpected numeric result.

Independence Across Slots

Distinct slots may, in principle, receive input elements that are algebraically related (for example, the same vector supplied to two different slots of the same space), but the multilinear structure treats each slot's input element as an independent variable for the purposes of the linearity condition: varying the input element in one slot while holding the others fixed, even if those other elements happen to equal the varying one, must still respect slotwise linearity in isolation.

Role in Defining Symmetry

Symmetry and antisymmetry statements are conditions on how the output changes when input elements are exchanged between slots of matching variance, holding the multiset of supplied elements fixed while permuting which slot receives which element. This distinguishes a genuine symmetry property of $T$ from a coincidental equality that arises only for one particular choice of input elements.

Summary of Key Points

An input element is the concrete vector or covector substituted into a slot at evaluation time, distinct from the slot's fixed structural definition.
Input elements must belong to the vector space bound to their slot; type mismatches are undefined, not merely incorrect.
Slotwise linearity allows any input element to be expanded in a basis, reducing evaluation of the tensor to a finite weighted sum over its components.
Basis vectors serve as the canonical input elements whose evaluations directly define the tensor's components.
Symmetry properties compare outputs across permutations of which slot receives which input element, for a fixed set of supplied elements.