The Double Dual: A Space Looks Back at Itself

Dualizing twice

V* is a vector space, so it has its own dual: V** = (V*)*, the double dual. An element of V** is a measurement-of-measurements — a linear functional that eats covectors and returns scalars. In Guide 2 we saw V ≅ V* but only after picking a basis. The double dual changes the story completely.

It is tempting to think 'dualize, dualize again, surely you just end up back where you started for the same boring reason'. Not quite. Dualizing once gave a dual space tied to V only through a basis. Dualizing a second time recovers V — but, as we'll see, through a map that needs no choices at all. The difference between those two trips is the whole point.

The evaluation map needs no basis

Here is the magic. Each vector v in V already knows how to act on functionals: just feed v to them. Define eval_v(f) = f(v). For fixed v this is linear in f, so eval_v is an element of V. The map v -> eval_v is the [[canonical-evaluation-map|evaluation map]] — and you wrote it down without ever choosing a basis. It is canonical**: defined by the structure alone.

V = R^2.  Vector v = (3, 5).
A covector f = [a  b] acts by f(v) = 3a + 5b.

Define eval_v : V* -> R  by  eval_v(f) = f(v) = 3a + 5b.
This is linear in (a,b), so eval_v lives in V**.

No basis was chosen to write eval_v down — only the
rule 'apply f to v'.  Swap a different vector w=(1,0):
  eval_w(f) = 1*a + 0*b = a   (it just reads off a).
Distinct vectors give distinct elements of V**  => map is injective.

The natural isomorphism and reflexivity

1. Injective: if v ≠ 0, some functional f has f(v) ≠ 0, so eval_v ≠ 0. No vector is invisible to all measurements.
2. Dimensions match: dim V** = dim V* = dim V (finite-dimensional), so injective forces surjective.
3. Therefore v -> eval_v is an isomorphism V ≅ V**, and it is canonical — the same for every basis.

A space for which this canonical map is an isomorphism is called reflexive. Every finite-dimensional space is reflexive, so we routinely identify V with V** and stop writing eval. The deep lesson: V and V* are genuinely different (row vs column), but a space and its double dual are the same thing wearing different clothes — and that sameness costs no choices.