4.6.1 Tensor Slotwise Additivity Property

Tensor Slotwise Additivity Property is the half of the multilinearity condition asserting that a tensor's defining map, when one argument slot is written as a sum of two vectors and every other slot is held fixed, evaluates to the sum of the map applied to each summand separately. It is the "sum-preserving" component of slotwise linearity, distinct from and complementary to homogeneity, and it alone is responsible for the ability to expand a tensor evaluation term by term whenever an argument is presented as a combination of others.

Formal Statement

The Additivity Condition

For a multilinear map $T : V_1 \times \cdots \times V_k \to F$, slotwise additivity requires that for every slot index $i$ and every pair $u, w \in V_i$,

with the arguments in every other slot fixed and identical on both sides of the equation. This condition must hold independently at every one of the $k$ slots, and each occurrence is checked with all other slots held constant.

Independence from Homogeneity

Additivity does not by itself imply homogeneity: a map could in principle satisfy additivity ($T(u+w) = T(u) + T(w)$ in each slot) over the rational numbers without extending to scalar multiplication by arbitrary elements of the field. The two conditions are logically separate, and both must be verified independently, though over most fields of interest in tensor algebra, such as the real or complex numbers, additivity together with a mild continuity or measurability assumption is enough to force homogeneity as well.

Consequences of Additivity Alone

Expansion Over Sums of Basis Vectors

If a vector in slot $i$ is written as a finite sum $v = \sum_j c_j e_j$, additivity in that slot (applied repeatedly) expands the evaluation into a sum of $k$-tuples, one term for each basis vector $e_j$ appearing in the decomposition, before homogeneity is even invoked to pull the coefficients $c_j$ outside:

Behavior Under Repeated Application

Additivity extends by induction from two summands to any finite number: if a slot's argument is a sum of $n$ vectors, the evaluation splits into $n$ separate terms, each obtained by substituting one summand while treating the rest of that slot's contribution as absent. This inductive extension is what makes additivity practically usable in symbolic manipulation of tensor expressions.

Role Within the Broader Multilinearity Property

One of Two Required Conditions

Slotwise additivity, together with slotwise homogeneity, jointly constitute the full slotwise linearity property that defines what it means for a map to be a tensor. Neither condition alone is sufficient: additivity without homogeneity would allow inconsistent scaling behavior, while homogeneity without additivity would allow inconsistent behavior under vector sums, and both failures would break the finite-component expansion that makes tensors computationally tractable.

Testing Additivity in Practice

To verify additivity for a candidate multilinear map, it suffices to check the additive identity for arbitrary pairs of vectors in one slot at a time, holding all other slots at arbitrary but fixed values, and to repeat this check independently for every slot in the argument structure; a single failure in any one slot disqualifies the entire map from being multilinear.

Summary of Key Points

• Slotwise additivity requires that summing two vectors in one slot, with all other slots fixed, produces an output equal to the sum of the two individual evaluations.
• It must be checked independently in every slot of the tensor's argument structure.
• It is logically distinct from slotwise homogeneity, though the two together constitute full slotwise linearity.
• Additivity is what permits a tensor evaluation to be expanded term by term whenever an argument is a sum, prior to any scalar factoring.
• It extends by induction to sums of any finite number of vectors within a single slot.