4.13.3 Tensor General Slot Additivity

Tensor General Slot Additivity is the uniform statement of the tensor multilinear additivity property across an arbitrary, unspecified slot of a tensor, expressing additivity as a single condition indexed by slot position rather than as a separate identity written out for each individual slot such as the first or the second. It packages the pattern observed in tensor first slot additivity and tensor second slot additivity, and in every other individual slot, into one general schema, making explicit that additivity is required uniformly across all p + q slots of a tensor, with no slot exempted or treated differently in kind.

The General Schema

Additivity Indexed by an Arbitrary Slot

For a type (p, q) tensor T on a vector space V, let k range over any of the p + q slots. General slot additivity states that for any two admissible arguments u and w of the appropriate kind for slot k, and any fixed choice of every other argument,

with the notation understood to place u + w, then u, then w, in slot k specifically, while every other slot, indicated by the placeholders ⋯, remains fixed and identical across all three terms. The index k is not fixed in advance but ranges over every one of the p + q positions, so general slot additivity is really a family of p + q individual statements, one for each value of k, unified under a single schema.

Quantifying Over All Slots at Once

General slot additivity can be stated compactly as the requirement that, for every k from 1 to p + q, the map obtained by fixing all slots except the k-th is additive in its one remaining argument; this universally quantified form is what distinguishes the general property from any single slot-specific instance such as first slot additivity, which fixes k = 1 outright.

Relation to the Specific Slot Instances

Recovering First and Second Slot Additivity as Special Cases

Setting k = 1 in the general schema recovers exactly the identity described by tensor first slot additivity, and setting k = 2 recovers exactly the identity described by tensor second slot additivity; every specific slot instance of additivity is obtained from the general schema by substituting a particular value for k, so the general property does not add any new mathematical content beyond what is already present in the collection of all individual slot instances, but it does express their common pattern as a single statement.

No Slot Is Structurally Distinguished

The general schema makes explicit that additivity does not privilege any particular slot: whatever holds for the first or second slot must equally hold for the third, the last, or any slot in between, with the only difference between slots being whether they are contravariant or covariant, which determines the kind of argument, vector or covector, that additivity is stated for at that position.

Why the General Form Is Needed

Tensors of Arbitrarily High Rank

For a tensor of low rank, such as a bilinear form with only two slots, it is practical to write out first slot additivity and second slot additivity as two separate identities; for a tensor of rank p + q much larger than two, writing out a separate additivity identity for every individual slot becomes unwieldy, and the general schema, indexed by an arbitrary k, is the practical way to state the full requirement without enumerating every slot by name.

Uniformity Required by the Definition of Multilinearity

The definition of a multilinear map requires additivity, together with homogeneity, in every slot without exception; general slot additivity is the precise formal expression of this "every slot without exception" requirement, ensuring that no proof or verification of multilinearity can be considered complete while omitting any slot from consideration.

Consequences of the General Property

Expansion Across Multiple Slots Simultaneously

Applying general slot additivity to several slots in succession, each slot's argument expanded as a sum of two or more vectors, produces a fully expanded sum with one term for every combination of summands chosen across the slots involved; this repeated application, licensed by general slot additivity holding at every slot, is what underlies the full expansion of a tensor evaluation when several arguments are simultaneously written as basis expansions.

Foundation for the Component Evaluation Formula

The tensor multilinear component evaluation formula, which expresses the evaluation of T as a sum over all combinations of basis indices, relies on general slot additivity holding at every one of the p + q slots simultaneously; without additivity guaranteed uniformly across every slot, the decomposition of each argument into a sum of scaled basis vectors could not be distributed through T to produce the component formula.

Diagrammatic Summary

The diagram shows every slot of the tensor, from the first to the last, subject to the same additivity pattern, with the dashed box marking an arbitrary slot k to emphasize that the property holds uniformly no matter which slot is singled out.