Editing Dual space (section)

=== Injection into the double-dual ===
There is a [[natural transformation|natural]] [[linear map|homomorphism]] <math>\Psi</math> from <math>V</math> into the double dual <math>V^{**}=\hom (V^*, F)</math>, defined by <math>(\Psi(v))(\varphi)=\varphi(v)</math> for all <math>v\in V, \varphi\in V^*</math>.  In other words, if <math>\mathrm{ev}_v:V^*\to F</math> is the evaluation map defined by <math>\varphi \mapsto \varphi(v)</math>, then <math>\Psi: V \to V^{**}</math> is defined as the map <math>v\mapsto\mathrm{ev}_v</math>.  This map <math>\Psi</math> is always [[injective]];<ref group=nb name="choice"/> and it is always an [[isomorphism]] if <math>V</math> is finite-dimensional.<ref>{{Harvp|Halmos|1974}} pp. 25, 28</ref>
Indeed, the isomorphism of a finite-dimensional vector space with its double dual is an archetypal example of a [[natural isomorphism]].
Infinite-dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.