4.1 Tensor Multilinear Map Structure

Tensor Multilinear Map Structure is the algebraic framework describing how a tensor of type (p, q) on a vector space V corresponds to a multilinear map that takes p covectors and q vectors (or, depending on convention, p vectors and q covectors) as arguments and returns a scalar, doing so linearly and separately in each of its arguments. This correspondence is the operational heart of tensor algebra: it converts an abstract element of a tensor product space into a concrete function-like object that can be evaluated, composed, contracted, and pulled back or pushed forward along linear maps.

Foundational Definition

Multilinearity

A map

with p copies of the dual space V* and q copies of V is called multilinear if, for every fixed choice of all arguments except one, the resulting function of that single remaining argument is linear. Explicitly, for the i-th vector slot,

for all scalars a, b and vectors v, w, with the same property holding independently in each covector slot.

Tensors as Multilinear Maps

The type (p, q) tensor space is canonically isomorphic to the space of multilinear maps described above, via the universal property of the tensor product. A simple tensor

acts on arguments by evaluating each covector-typed argument on the corresponding vector factor and each vector-typed argument against the corresponding covector factor, then multiplying all the resulting scalars together.

Coordinate Description

Components as Evaluations on a Basis

Fixing a basis {e_1, ..., e_n} of V and dual basis {e^1, ..., e^n} of V*, the components of a (p, q) tensor T are obtained by evaluating the multilinear map on basis vectors and basis covectors:

Multilinearity guarantees these n^(p+q) scalars determine T completely: for arbitrary arguments expanded in the basis, the value of T is obtained by substituting components and summing over repeated indices, which is precisely the Einstein summation convention.

General Evaluation Formula

For arbitrary vectors v_k = v_k^i e_i and covectors ω^l = ω^l_j e^j,

This shows explicitly that the multilinear map structure and the component array picture are two faces of the same object.

Structural Properties

Symmetry and Antisymmetry

A multilinear map's structure can be further classified by its behavior under permutation of arguments of the same type. If T is invariant under every permutation of its vector arguments, it is symmetric; if it changes sign under every transposition, it is antisymmetric (alternating). These subspaces of the full multilinear map space give rise, respectively, to symmetric tensors and to differential-form-like alternating tensors.

Multilinearity and Rank

The pair (p, q) is the valence or rank of the tensor and equals the number of independent argument slots in its multilinear map presentation. Contraction reduces valence by pairing one vector slot with one covector slot and summing over the paired index, effectively removing one V and one V* factor from the multilinear signature.

Compatibility with Linear Maps

If f: V → W is linear, the multilinear map structure transforms functorially: covector slots pull back contravariantly via the dual map f*, while vector slots push forward covariantly via f, provided f is invertible or the tensor is purely covariant. This functoriality is what allows tensors, understood as multilinear maps, to be transported consistently between vector spaces connected by linear or smooth maps.

Operations Built on the Multilinear Map Structure

Tensor Product

Given multilinear maps T of type (p, q) and S of type (p', q'), their tensor product T ⊗ S is the multilinear map of type (p + p', q + q') formed by evaluating T on the first block of arguments and S on the second block, then multiplying the results. This operation is associative and respects the multilinear structure of each factor independently.

Composition with Multilinear Substitution

Multilinear maps compose with linear substitutions in each slot: replacing an argument by a linear combination of basis elements before evaluation is equivalent to contracting the tensor's index with the coefficients of that combination, which is the operational basis for change-of-basis formulas and for pullback and pushforward computations on tensors of arbitrary valence.