4.7.2 Tensor Multiple Vector Space Domain

Tensor Multiple Vector Space Domain is the generalized version of a tensor's domain structure in which the factor spaces are not required to all coincide with a single space $V$ or its dual $V^{}$, but may instead be several genuinely distinct vector spaces $V_1, \ldots, V_k$, possibly of different dimensions, combined via a Cartesian product to serve as the input to a multilinear map. It broadens the ordinary notion of a tensor built purely from copies of $V$ and $V^{}$ into the wider setting of multilinear algebra over arbitrarily many, arbitrarily different vector spaces.

Formal Definition

The General Multi-Space Product

For distinct vector spaces $V_1, \ldots, V_k$, possibly over the same field $F$ but with no requirement that any two coincide or share a dimension, the multiple vector space domain is

A multilinear map $T$ on this domain is required only to be linear in each slot separately, exactly as in the single-space case, but there is no assumption that swapping two arguments even makes sense, since arguments from different slots may come from entirely unrelated spaces with no canonical way to compare them.

Specialization to the Ordinary Tensor Case

The familiar notion of a type $(r, s)$ tensor over a single space $V$ is the special case of a multiple vector space domain in which $r$ of the factors are set equal to $V^{*}$ and $s$ of the factors are set equal to $V$. The general multiple vector space domain removes this restriction, allowing each factor to be an independent space with its own dimension and its own basis.

Consequences of Allowing Distinct Factor Spaces

Loss of a Universal Symmetry Concept

When all factors coincide, permuting two slots is meaningful because both slots draw from the same space, and this is what allows symmetric and antisymmetric tensors to be defined. When the factor spaces are genuinely distinct, exchanging the arguments of two different-dimensional slots is not even well-typed in general, so notions of symmetry and antisymmetry only apply among subsets of slots that happen to share the same factor space.

Component Count Reflects All Factor Dimensions

If each $V_i$ has dimension $n_i$, the total number of independent components of a multilinear map on this domain is the product $n_1 n_2 \cdots n_k$, generalizing the formula $n^{r+s}$ that applies when every factor shares a common dimension $n$. Distinct factor dimensions mean the component count can grow unevenly depending on which slots carry larger spaces.

Contraction Requires Compatible Spaces

Contraction, which pairs a slot bound to $V_i$ with a slot bound to $V_i^{*}$ (its dual), remains meaningful only between a slot and the dual of the exact same space; two slots bound to two different, unrelated spaces $V_i$ and $V_j$ cannot be contracted unless an additional isomorphism between them is separately supplied.

Applications of the Multiple Vector Space Domain

Bilinear and Multilinear Maps Between Different Spaces

The most familiar example is a bilinear map $B : V \times W \to F$ between two genuinely different spaces $V$ and $W$, such as a pairing used to define a tensor product $V \otimes W$ itself; this is the arity-2 case of a multiple vector space domain with no requirement that $V$ and $W$ coincide.

Building Blocks for General Tensor Products

The tensor product of several different vector spaces, $V_1 \otimes \cdots \otimes V_k$, is constructed as the universal linearization of exactly this kind of multiple vector space domain, generalizing the single-space tensor product construction to accommodate inputs of different types and dimensions.

Multilinear Algebra Over Heterogeneous Structures

Constructions such as multilinear maps that combine a vector space with a space of matrices, or a space of functions with a space of coefficients, rely on the multiple vector space domain framework, since these underlying spaces have no natural reason to be identified with one another or with a common base space $V$.

Summary of Key Points

• A multiple vector space domain generalizes the ordinary tensor domain by allowing each factor to be a distinct vector space rather than copies of a single $V$ and its dual.
• The ordinary type $(r,s)$ tensor domain is recovered as the special case where the factors are restricted to $V$ and $V^{*}$.
• Symmetry concepts apply only among slots sharing the same factor space, since exchanging arguments across unrelated spaces is not well-typed.
• The total component count is the product of the individual factor dimensions, which need not be equal across slots.
• Contraction requires a slot bound to a space and another bound to its exact dual, restricting which slot pairs can be paired without extra structure.