14.23.2 Tensor Map Product Linear Map Boundary

Tensor Map Product Linear Map Boundary is the precise delineation of which properties a map must satisfy for the tensor product of maps construction to apply to it at all, marking the edge past which semilinear maps, affine maps, and maps defined over mismatched base fields each require a modified construction or admit no analogous tensor product in the ordinary sense.

The Precise Requirement

Both Additivity and Homogeneity Are Needed

The tensor product of maps construction requires $f$ and $g$ to be linear over the same field $K$, meaning both additive, $f(v_1+v_2)=f\left(v_1\right)+f\left(v_2\right)$, and homogeneous with respect to $K$-scalar multiplication, $f\left(\alpha v\right)=\alpha f\left(v\right)$ for every scalar $\alpha \in K$; both properties are used directly in the domain verification step that lets the bilinear assignment $(v,w)\mapsto f\left(v\right)\otimes g\left(w\right)$ descend to a well-defined map on $V\otimes W$.

Neither Property Alone Suffices

A map satisfying additivity without homogeneity, or homogeneity without additivity, does not respect the full set of relations generated in the construction of the tensor product, so domain verification fails at the corresponding step, and no linear tensor product map is produced even though the map may still send the relevant simple tensors somewhere sensible on an individual basis.

Semilinear Maps

The Boundary Case

A semilinear, or conjugate-linear, map $f:V\to V^{\prime }$ over a field with an involution, most commonly complex conjugation on $ℂ$, satisfies additivity but replaces homogeneity with $f\left(\alpha v\right)=\bar{\alpha }f\left(v\right)$, using the conjugate scalar instead.

Why the Ordinary Construction Fails

Because the tensor product's defining relation $(\alpha v,w)\equiv (v,\alpha w)$ is built using ordinary, unconjugated scalar multiplication, a semilinear map does not respect this relation in the ordinary sense, and the assignment $(v,w)\mapsto f\left(v\right)\otimes g\left(w\right)$ fails domain verification when $f$ or $g$ is semilinear rather than linear; a separate construction, tracking which factors are conjugated, is required instead, and the resulting object is not the ordinary tensor product of maps described by this theory.

Affine Maps

The Boundary Case

An affine map $f\left(v\right)=\alpha \left(v\right)+v_0$, with $\alpha$ linear and $v_0$ a fixed nonzero vector, fails both additivity and homogeneity in general.

Why No Analogous Construction Exists at This Boundary

Since affine maps do not preserve the zero vector, $f\left(0\right)\ne 0$ in general, no version of the tensor product construction applies to them directly; the tensor product of maps is intrinsically tied to the linear structure of vector spaces, and affine maps belong properly to affine space theory, a related but distinct setting in which the analogous combination of two affine maps, if defined at all, requires an entirely different construction built on affine rather than tensor products.

Maps Over Mismatched Base Fields

The Boundary Case

If $f:V\to V^{\prime }$ is linear over a field $K$ and $g:W\to W^{\prime }$ is linear over a different field $L$, with no common field over which both maps are simultaneously linear, the tensor product of maps as defined here does not apply, since the vector spaces $V\otimes W$ and $V^{\prime }\otimes W^{\prime }$ are not even canonically defined without first fixing a single common field to tensor over.

Resolution by Restriction or Extension of Scalars

This boundary is typically crossed, not by modifying the tensor product construction itself, but by first applying restriction of scalars, viewing $g$ as linear over a smaller field common to both, or extension of scalars, enlarging $V$ and $V^{\prime }$ to vector spaces over a common larger field; only after this preliminary adjustment does the ordinary tensor map product construction become applicable, and the boundary itself marks the point at which this adjustment becomes necessary rather than optional.

Summary of the Linear Map Boundary

A Single Requirement With Several Distinct Failure Modes

Every case discussed here traces back to the same underlying requirement, linearity over a single, common field, failing in a different specific way: semilinear maps fail homogeneity in the ordinary sense, affine maps fail both additivity and the preservation of zero, and maps over mismatched fields fail to share a common scalar structure at all; recognizing which of these three failure modes is present is what determines which alternative construction, if any, is appropriate once the ordinary tensor product of maps no longer applies.