4.24.2 Tensor Bilinear Map Boundary

Tensor Bilinear Map Boundary is the delineation of the two-argument case of the multilinear map boundary, fixing exactly which maps qualify as bilinear maps for the purpose of tensor identification: functions of two vector arguments that are linear in each argument separately when the other is held fixed, taking values in the base field or a designated target vector space. As the k = 2 instance of multilinearity, the bilinear boundary is the first level at which the distinction between separate linearity and joint linearity becomes visible, since a bilinear map is essentially never linear as a function of the pair (u, v) treated as a single vector.

The Defining Condition

Linearity in Each Slot

A map

lies inside the boundary when, for fixed v, the map u ↦ B(u, v) is linear, and for fixed u, the map v ↦ B(u, v) is linear. Written out, this requires

together with the matching additivity and homogeneity conditions in the second slot. Both conditions must hold independently; satisfying linearity in the first argument alone, or in the second alone, places a map only inside the linear boundary applied to one slot, not inside the bilinear boundary.

Non-Joint-Linearity as a Structural Feature

A bilinear map is not linear on V × V regarded as a vector space in its own right, since

expands into four separate terms rather than the two that joint linearity would require. This expansion into cross terms is a defining structural feature of the boundary, not a defect: it is exactly what allows a bilinear map to correspond to a tensor of type (2, 0), since the four-term expansion mirrors how such a tensor acts on a sum of decomposable elements.

What Falls Outside the Boundary

Symmetric Constructions That Break Linearity

A map such as B(u, v) = ⟨u, v⟩², the square of a bilinear form, fails linearity in each slot despite being built from a genuinely bilinear ingredient, because squaring destroys additivity; quadratic forms of this kind sit outside the bilinear boundary and belong instead to the separate framework of quadratic forms.

Maps Linear in Only One Argument

A map linear in u for fixed v but only continuous, or only positively homogeneous, in v for fixed u fails the boundary condition on the second slot and is excluded in full, even though half of the required structure is present.

Relation to Adjacent Boundaries

Between the Linear and General Multilinear Boundaries

Tensor Bilinear Map Boundary occupies the position between Tensor Linear Map Boundary and the general Tensor Multilinear Map Boundary: every bilinear map decomposes, slot by slot, into linear maps once the other argument is fixed, and every bilinear map is simultaneously a multilinear map with k = 2. No additional condition beyond separate linearity in two slots is imposed at this level, so the bilinear boundary inherits its content entirely from the general multilinear condition specialized to k = 2.

Source of Bilinear Forms and the Tensor Product

Bilinear maps with codomain F are exactly the bilinear forms, and it is precisely this boundary that the universal property of the tensor product is stated against in its most classical form: a bilinear map out of V × V factors uniquely through V ⊗ V. Maps outside the bilinear boundary admit no such factorization, since the universal property applies only to maps already known to be bilinear.