4.3 Tensor Trilinear Map Structure

Tensor Trilinear Map Structure is the framework describing functions of three vector-space arguments that are linear in each argument separately while the other two are held fixed, generalizing bilinear maps by adding a third independently linear slot. Formally a trilinear map is a function $$T:V\times W\times X\to U$$ satisfying linearity in the first argument when the second and third are fixed, linearity in the second argument when the first and third are fixed, and linearity in the third argument when the first and second are fixed. This structure is the natural next step after bilinear maps in the hierarchy of multilinear maps used to construct and analyze tensors of rank three.

Defining Conditions

Slotwise Linearity

For fixed $$w\in W$$ and $$x\in X$$, the map $$v\mapsto T(v,w,x)$$ must be linear, and analogous statements hold when the first-and-third or first-and-second arguments are frozen. Explicitly, for the first slot:

with the identical form of condition required for the second and third slots. All three conditions must hold simultaneously; a function linear in each pair of slots but not in the third would not qualify as trilinear.

Non-Linearity Over the Product Space

Just as with bilinear maps, a trilinear map is not linear when $$V\times W\times X$$ is regarded as a single vector space, since scaling all three arguments simultaneously by $$\alpha$$ scales the output by $${\alpha }^3$$ rather than by $$\alpha$$. Trilinearity is exclusively a statement about independent linearity in each separate slot.

Coordinate Representation

Four-Index Structure Constants

With bases $$\{e_i\}$$, $$\{f_j\}$$, $$\{g_k\}$$ on $$V$$, $$W$$, $$X$$ and $$\{u_l\}$$ on $$U$$, a trilinear map is captured entirely by the four-index array

and general values are recovered by

When the output space is the scalar field, the single output index disappears and the map reduces to a rank-three covariant tensor with components $$c_{ijk}$$.

Relation to the Triple Tensor Product

Universal Property

The universal property of the tensor product extends from two to three factors: every trilinear map $$T:V\times W\times X\to U$$ factors uniquely through a linear map $$\tilde{T}:V\otimes W\otimes X\to U$$ such that

This is what makes the triple tensor product the universal recipient of trilinear maps, mirroring the bilinear case one dimension higher.

Iterated Currying

A trilinear map can be built by iterating bilinear currying: fixing the third argument gives a bilinear map in the first two, which factors through $$V\otimes W$$, and the resulting family of bilinear maps parametrized by $$x\in X$$ is itself linear in $$x$$, showing that trilinear maps on $$V\times W\times X$$ correspond to linear maps on $$(V\otimes W)\otimes X$$, which is naturally isomorphic to $$V\otimes W\otimes X$$.

Symmetry Types

Fully Symmetric and Alternating Cases

When $$V=W=X$$, a trilinear map may be classified by its behavior under permutation of arguments. A fully symmetric trilinear form satisfies $$T(v,w,x)=T(\sigma (v,w,x))$$ for every permutation $$\sigma$$ of the three slots, while a fully alternating one changes sign under any transposition, generalizing skew-symmetric bilinear forms to the determinant-like structure that underlies the scalar triple product.

Mixed Symmetry

More generally a trilinear map may be symmetric in one pair of slots and unrelated in behavior to the third, reflecting the partial symmetries that appear in tensors such as the Riemann curvature tensor's index pairs, a phenomenon with no analogue in the purely bilinear case since two slots offer only the single symmetric-versus-antisymmetric dichotomy.

Role in Constructing Higher Tensors

Trilinear map structure is the mechanism by which rank-three tensors are both defined and manipulated: every rank-three tensor over finite-dimensional spaces corresponds to exactly one trilinear map on the appropriate product of vector and dual spaces, and operations such as contraction of one index reduce a trilinear map to a bilinear one by evaluating one slot against a fixed dual vector, linking trilinear structure directly back to the bilinear framework from which it is built.