I have a problem where im asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is bijective.

I showed easily that it is injective, now i have to show it is surjective.

I came up with this but i need someone to tell me how wrong or close to right i am:

To prove F is surjective, i claim that for any element of X*, call it t, there is an element b in X. Since X is a complete metric space then X=X*, so t=b

Thus t is the element in X such that F(t)=t.

Any guidence will be apreciated