Riesz Representation Theorem in Wikipedia vs. Rudin's RCA

Question

In Rudin's Real & Complex Analysis theorem 2.14, the Riesz representation theorem gives (in my very rough phrasing) an injection from linear functionals on a space to positive Borel measures which represent the functionals, in a sense.

In Wikipedia's article about said theorem, however, the correspondence is stated between continuous linear functionals on a Hilbert space and the Hilbert space itself.

Trying to reconcile the two, there must be then a one-to-one isomorphism between the set of measures and the Hilbert space: but what is it? Does every point in the Hilbert space define a measure and vice versa?

Answer 1

As mentioned in comments, there are different "Riesz representation theorems".

• The Hilbert space version (pretty easy to prove) gives a natural correspondence between bounded linear functionals and the elements of the Hilbert space.
• The representation of linear functionals on the spaces $C(K)$ (continuous functions on a compact space), $C_c(X)$ (continuous with compact support) and $C_0(X)$ (continuous and vanishing at infinity) involves measures and is considerably harder.