2.17.3 Tensor Real Multilinear Operation

A tensor real multilinear operation combines multiple vectors into a scalar through a bilinear, associative, and distributive process.

Tensor Real Multilinear Operation is a function of several arguments drawn from a real vector space V and its dual V* that is linear in each argument separately with respect to real scalars and real vector addition, holding all other arguments fixed. It is the operational core of tensor algebra over the reals: every tensor is, by definition, either such a multilinear operation itself or an element of the tensor product space that classifies such operations, and every algebraic manipulation performed on tensors — sums, scalar multiples, tensor products, contractions — must preserve this multilinearity to remain well defined.

Formal Characterization

Linearity in Each Slot

A real multilinear operation on V of degree k is a map

T : V \times V \times \dots \times V \to R

with k copies of V, such that for every index m and every fixed choice of the remaining k - 1 arguments, the induced function of the m-th argument alone is linear over R:

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

for all real scalars a, b and vectors u, w ∈ V. This must hold in every argument slot independently, which is a strictly stronger condition than joint linearity of all arguments considered together.

Distinction From a Single Linear Map

A real multilinear operation of degree 1 reduces to an ordinary linear functional on V, an element of V*. Multilinearity only becomes a distinct notion starting at degree 2, where the operation is linear in each of two arguments but generally not linear in the pair (v, w) treated as a single vector in V ⊕ V, since, for instance, T(v + v', w + w') does not generally equal T(v, w) + T(v', w') unless cross terms vanish.

The Multilinear Map as the Definition of a Tensor

Type (p, q) Tensors as Multilinear Operations

A real type (p, q) tensor is exactly a real multilinear operation taking p covector arguments from V* and q vector arguments from V and returning a real scalar. Real multilinearity in each of the p + q slots is the defining property that must be checked to confirm that a proposed rule actually defines a tensor.

Multilinearity and the Universal Property of the Tensor Product

The tensor product space V ⊗ W is constructed so that real bilinear (and, more generally, multilinear) operations out of V × W correspond bijectively to real linear maps out of V ⊗ W. This universal property is what allows every multilinear operation to be re-expressed as a single linear map on a larger space, unifying the multilinear and linear points of view.

Bilinear maps (V \times W \to R) ≅ Linear maps (V \otimes W \to R)

Operations That Preserve Multilinearity

Real Linear Combinations of Multilinear Operations

If S and T are real multilinear operations of the same degree on V, then aS + bT, for real scalars a, b, is again a real multilinear operation of that degree, since linearity in each slot is preserved under real linear combination. This is what makes the set of degree-k multilinear operations itself a real vector space.

Composition With Linear Maps

If f : V → V is a real linear map and T is a real multilinear operation of degree k, then the pullback (f^*T)(v_1, ..., v_k) = T(f(v_1), ..., f(v_k)) is again a real multilinear operation, since composing a linear map with a linear function of one variable preserves linearity in that variable. This construction is the basis for how tensors transform under linear changes of coordinates.

Symmetrization and Antisymmetrization

Given a real multilinear operation T of degree k, averaging T over all permutations of its arguments (with a sign for antisymmetrization) produces new multilinear operations that are respectively fully symmetric or fully antisymmetric. Both operations remain real multilinear operations because permuting arguments and taking real linear combinations of the results preserve linearity in each slot.

Why Real Coefficients Matter Here

No Sesquilinear Ambiguity

Because the scalars are real, "linear in each argument" is an unambiguous single notion: there is no distinction, as there is over the complex numbers, between a slot that is linear and one that is conjugate-linear. Every real multilinear operation is linear, full stop, in every one of its arguments.

Compatibility With the Real Coordinate System

When expressed in a real coordinate system, a degree-k real multilinear operation is completely determined by its values on all k-tuples of basis vectors, giving n^k real numbers, since multilinearity allows any argument to be expanded in the basis and the operation evaluated term by term using real distributivity.

Diagrammatic Summary

Each incoming arrow into T represents a slot in which the operation is linear over R independently of the values placed in every other slot, which is the essential content of real multilinearity.