3.15.4 Tensor Dual Map Composition Behavior

Tensor dual maps compose in a way that reflects the structure of their underlying spaces, revealing key properties in linear algebra and functional analysis.

Tensor Dual Map Composition Behavior is the set of rules governing how the dual map construction interacts with composition of linear maps, most notably the contravariant identity that dualizing a composite reverses the order in which the two maps are combined. If f: V → W and g: W → U are linear maps, their composite g ∘ f: V → U has a dual (g ∘ f)*: U* → V* that equals f* ∘ g* rather than g* ∘ f*, and this order-reversal, together with the way composition behaves on tensor products and iterated compositions, forms the full composition behavior of the dual map construction.

The Order-Reversal Law

Statement of the Law

For composable linear maps f: V → W and g: W → U, the dual of the composite satisfies

(g \circ f)^{*} = f^{*} \circ g^{*}

Both sides are maps from U* to V*, so the equation is meaningful as an identity between two linear maps with the same domain and codomain, and it asserts that they agree on every input.

Proof from the Defining Property

The law follows directly from the definition of the dual map through precomposition. For any φ ∈ U* and v ∈ V,

((g \circ f)^{*} (φ)) (v) = φ (g (f (v)))

and computing the right-hand side of the law on the same inputs gives (f*(g*(φ)))(v) = (g*(φ))(f(v)) = φ(g(f(v))), the identical expression. Since the two sides agree on every φ and every v, the maps themselves are equal.

Why Reversal, Not Preservation

If dualization preserved order, one would expect (g ∘ f)* = g* ∘ f*, but this expression is not even well-defined in general: g* maps U* → W* and f* maps W* → V*, so g* ∘ f* would require the codomain of f* to match the domain of g*, which it does not, since f* lands in V* while g* expects input from U*. Only the reversed order f* ∘ g* is composable, which is itself evidence that reversal is forced by the domain and codomain structure of the dual maps involved.

Consequences for Iterated Composition

Chains of Three or More Maps

For a chain f_1: V_0 → V_1, f_2: V_1 → V_2, ..., f_k: V_{k-1} → V_k, the dual of the full composite reverses the entire order:

(f_{k} \circ \dots \circ f_{1})^{*} = f_{1}^{*} \circ \dots \circ f_{k}^{*}

The dual of a chain is the chain of duals applied in the exact opposite sequence, which is precisely the statement that duality is a contravariant functor on the category of vector spaces.

Composition of an Endomorphism with Itself

When f: V → V is an endomorphism composed with itself k times, the order-reversal law still applies, but since every map involved is f, the reversal has no visible effect on which map appears where:

(f^{k})^{*} = (f^{*})^{k}

The dual of the k-th power of f equals the k-th power of the dual of f, since reversing a sequence made of identical entries yields the same sequence.

Composition Behavior in Matrix Terms

Transpose of a Product

The order-reversal law is the abstract counterpart of the elementary matrix identity (AB)^T = B^T A^T. If f and g are represented by matrices A and B respectively, with the product BA representing g ∘ f in the appropriate bases, then the dual composite f* ∘ g* is represented by A^T B^T, matching (BA)^T exactly. Composition behavior at the abstract level and transpose behavior at the matrix level are the same statement in two languages.

Preservation of Identity and Inverses

Composition behavior interacts cleanly with identities and inverses: (id_V)* = id_{V*}, and if f is invertible, (f^{-1})* = (f*)^{-1}. This follows by applying the order-reversal law to f ∘ f^{-1} = id_V and f^{-1} ∘ f = id_V, dualizing both sides, and observing that the resulting equations identify (f*)^{-1} with (f^{-1})*.

Composition Behavior on Tensor Products

Dual of a Tensor Product of Composites

If f_1 ∘ g_1: V → U and f_2 ∘ g_2: V' → U' are two composites, the dual of their tensor product (f_1 ∘ g_1) ⊗ (f_2 ∘ g_2) factors consistently with the order-reversal law applied to each tensor factor independently:

((f_{1} \circ g_{1}) \otimes (f_{2} \circ g_{2}))^{*} = (g_{1}^{*} \circ f_{1}^{*}) \otimes (g_{2}^{*} \circ f_{2}^{*})

Each tensor factor obeys the order-reversal law on its own, and the tensor product of the resulting duals matches the dual of the original tensor product, confirming that composition behavior is compatible with the tensor product construction.

Relevance to Tensor Transformation Chains

When a type (p, q) tensor is transformed by a sequence of endomorphisms applied one after another, composition behavior guarantees that transforming by the full composite in one step gives the same result as transforming by each map individually in reverse order on the covariant slots, while the contravariant slots follow the original order. This consistency is essential for tensor calculus to behave predictably under sequences of coordinate changes.

Diagrammatic Summary

The top row shows f followed by g carrying V to U through W. The bottom row shows the dual maps running in the reverse spatial order, with g* applied first and f* applied second, illustrating that (g ∘ f)* = f* ∘ g*.