4.19.5 Tensor Bilinear Form Pairing Relation

Tensor Bilinear Form Pairing Relation is the way a bilinear form f: V × W → F links two, possibly distinct, vector spaces V and W by producing a scalar from one vector out of each, thereby establishing a relationship between the two spaces that can be studied independently of any symmetry condition, since V and W need not coincide for a pairing to be defined.

Pairing as a General Relationship Between Two Spaces

Beyond Forms on a Single Space

While a bilinear form f: V × V → F on a single space admits questions of symmetry, a pairing f: V × W → F between two different spaces admits no such question, since swapping arguments across two different spaces is not meaningful. What a pairing does provide, regardless of whether V and W coincide, is a way for elements of V to probe elements of W, and vice versa, through the bilinear evaluation f(v, w).

The Induced Maps

A pairing f gives rise to two linear maps, one in each direction:

The pairing relation is understood equally well through f itself, through φ_f, or through ψ_f, since each determines the other two completely.

Perfect Pairings

Definition

A pairing is perfect if both induced maps φ_f: V → W* and ψ_f: W → V* are isomorphisms. In finite dimensions this requires dim(V) = dim(W) and non-degeneracy of f in both directions; a perfect pairing identifies each space with the dual of the other, making V and W interchangeable for the purposes of the pairing.

The Canonical Example

The evaluation pairing V* × V → F, (φ, v) ↦ φ(v), is perfect whenever V is finite-dimensional, since φ_f: V* → V* is the identity and ψ_f: V → V** is the canonical isomorphism identifying V with its double dual. This pairing is the archetype against which all other perfect pairings are compared, and any perfect pairing between V and W can be used to transport constructions native to V* over to W, and vice versa.

Non-Perfect Pairings and Partial Information

One-Sided Non-Degeneracy

A pairing can be non-degenerate on one side without being perfect: if dim(W) > dim(V), the map φ_f: V → W* can be injective (left non-degenerate) yet fail to be surjective, since its image cannot span the larger space W*; the pairing relation in this case conveys full information about V via its image in W*, but W retains directions invisible to any element of V under the pairing.

Pairings With a Radical

If f is degenerate on the V side, vectors in the radical of φ_f are indistinguishable from zero as far as the pairing with W is concerned; the pairing relation, restricted to V / rad(φ_f), still gives a well-defined, now left non-degenerate, relationship with W, isolating the part of V that the pairing actually detects.

Examples of Pairing Relations

Trace Pairing on Matrices

The pairing (A, B) ↦ tr(AB) between n × n matrices and themselves is perfect, identifying the space of matrices with its own dual and underlying the standard way linear functionals on matrix space are represented as "matrix dotted against another matrix."

Pairing Between a Space and Its Quotient's Dual

Given a subspace U ⊆ V, there is a natural pairing between V/U and the annihilator U° ⊆ V* (functionals vanishing on U), given by ([v], φ) ↦ φ(v), well defined since φ vanishes on U; this pairing is perfect in finite dimensions, since (V/U)* ≅ U° canonically, illustrating how pairing relations arise naturally between spaces built from different constructions rather than only between a space and a copy of itself.

Pairing Between Homology and Cohomology

In algebraic topology, cup product and evaluation give a pairing between cohomology groups and homology groups with coefficients in a field, which is perfect exactly when the universal coefficient theorem provides no extension term, illustrating pairing relations arising between spaces constructed by entirely different means, unified only by the bilinear relationship connecting them.

Pairing Relations and the Tensor Product

Pairings as Elements of a Dual Tensor Product

Every pairing f: V × W → F corresponds, via the universal property of the tensor product, to a linear functional on V ⊗ W, and in finite dimensions to an element of V* ⊗ W*. The rank of this tensor, the minimal number of elementary tensors needed to express it, matches the rank of f viewed as a matrix relative to any bases, connecting the abstract pairing relation directly to concrete tensor-rank considerations.

Perfect Pairings and Canonical Isomorphisms

A perfect pairing between V and W corresponds to an element of V* ⊗ W* of maximal rank equal to dim(V) = dim(W), and conversely such a maximal-rank tensor always defines a perfect pairing; this correspondence is the tensor-algebraic restatement of the fact that a perfect pairing is precisely a bilinear form whose associated matrix is square and invertible.

Why the Pairing Perspective Is Useful

Separating Structural Roles

Framing a bilinear form as a pairing relation between two spaces, rather than as a form on a single space, clarifies constructions where V and W genuinely play different roles, such as a space of functions paired against a space of measures, or a space of tangent vectors paired against a space of covectors, where imposing a false symmetry between the two sides would obscure rather than illuminate the underlying structure.

Guiding the Search for Isomorphisms

Recognizing a bilinear form as a candidate pairing relation directs attention to checking non-degeneracy in each direction separately, and to comparing the dimensions of V and W, before asking whether the pairing identifies the two spaces perfectly, rather than assuming any bilinear relationship automatically identifies its two arguments' spaces with each other's duals.