double dual

The double dual is the dual of the dual: (V*)*, the space of all linear functionals on V*. At first this sounds like an infinite regress — measurements of measurements of measurements — but something remarkable stops the spiral after one step.

There is a natural map from V into (V*)* that needs no choice of basis at all. A vector v becomes the functional 'evaluate me': it sends a functional f to the number f(v). In finite dimensions this map is an isomorphism, so V and (V*)* are not merely the same size — they are the SAME, canonically. The double dual is V back again, dressed differently.

Contrast this with the single dual. V is isomorphic to V*, but only after a basis choice; the isomorphism is arbitrary. The map V -> (V*)* uses no choices, which is why mathematicians call it canonical or natural. It is the cleanest example of a natural isomorphism in all of linear algebra.

Practical upshot: because V = (V*)* canonically, you may freely regard a vector as a functional on functionals. This symmetry powers the whole annihilator/transpose machinery and lets duality theorems run in both directions. In infinite dimensions the map still injects V into (V*)* but need not be onto — spaces where it is onto are exactly the reflexive ones.