14.5.5 Tensor Linear Functional Product Multilinear Role

Tensor Linear Functional Product Multilinear Role is the part played by tensor products of linear functionals in representing and generating multilinear maps, since every elementary tensor of functionals corresponds to a multilinear map of a particularly simple, factorizable form, and general multilinear maps are built as sums of such elementary products.

From Functional Products to Multilinear Maps

The Multilinear Map Associated to an Elementary Product

Given functionals phi-1 through phi-k on spaces V-1 through V-k respectively, the elementary tensor phi-1 tensor cdots tensor phi-k corresponds to the multilinear map

a map linear in each argument separately, since fixing all but one v-i reduces mu to a scalar multiple of phi-i, which is itself linear.

Multilinearity as a Consequence of the Product Structure

The multilinear role rests on the observation that a product of individually linear scalar functions is linear in each factor of the product separately whenever the other factors are held fixed, so multilinearity of mu is inherited directly from linearity of each phi-i rather than requiring any separate verification.

Generating General Multilinear Maps

Sums of Elementary Products

A general multilinear map on V-1 times cdots times V-k need not itself be expressible as a single elementary tensor of functionals, but in the finite-dimensional case every multilinear map is expressible as a finite sum of such elementary products, matching the fact that the tensor product of the dual spaces is spanned by elementary tensors of individual functionals.

Rank of a Multilinear Map

The minimal number of elementary products needed to express a given multilinear map as such a sum plays the role of a rank for that map, generalizing the rank of a bilinear form to the multilinear setting, with a multilinear map of rank one being exactly one expressible as a single elementary tensor of functionals.

Role in Coordinate Representations

Coordinates via Dual Basis Products

If dual bases are fixed for each V-i, the multilinear role of tensor products of functionals is realized concretely: the elementary products of dual basis vectors form a basis for the space of all multilinear maps on V-1 times cdots times V-k, and every multilinear map is represented, with respect to this basis, by a multi-index array of scalar coefficients.

Correspondence with the Coefficient Array

Under this correspondence, evaluating a multilinear map on a k-tuple of basis vectors recovers the corresponding entry of its coefficient array directly, and the multilinear role of elementary functional products is precisely to supply the basis elements against which these coefficients are measured.

Role in Symmetrization and Alternation

Symmetric Multilinear Maps

When all the spaces V-1 through V-k coincide with a single space V, the multilinear role of tensor products of functionals extends to producing symmetric multilinear maps by averaging an elementary product over all permutations of its arguments, connecting elementary tensors of functionals to the construction of symmetric tensors on the dual of V.

Alternating Multilinear Maps

Similarly, averaging an elementary product with alternating signs over all permutations produces an alternating multilinear map, connecting elementary tensors of functionals to the construction of alternating tensors, or exterior forms, on the dual of V, and showing that the multilinear role of functional products underlies both the symmetric and the alternating branches of multilinear algebra built from a single space.