Different versions of Riesz Theorems In Wikipedia, there are three versions of Riesz theorems:

1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space;
2 The representation theorem for positive linear functionals on $C_c(X)$, where $X$ is a locally compact Hausdorff space;
3 The representation theorem for the dual of $C_0(X)$, where $X$ is a locally compact Hausdorff space.

I was wondering

*

*if none of the three versions is
more general  than the others, in
the sense that no one can be derived from another?

*when two or three of them can coincide?

Thanks and regards!
 A: These are three different theorems and there's no relation between 1) and the others except in the case when $X$ is finite, where all three theorems coincide (since $\ell^2(X)$, $C_0(X)$ and $C_c(X)$ then are the same topological vector space). Note however that $C_0(X)$ and $C_c(X)$ are never Hilbert spaces (unless the locally compact space $X$ is empty or a point), so 2) and 3) can't have a direct relation to 1).
Since positivity implies continuity 2) can be interpreted as characterizing continuous linear functionals on $C_c{(X)}$ as well, and I'm addressing this version below.
The results 2) and 3) are closely related and often 3) is proved as a corollary of 2).
Note that the space $C_c(X)$ is dense in $C_{0}(X)$ with respect to the sup-norm. So a continuous linear functional on $C_c(X)$ (= a signed Radon measure) extends (uniquely) to a continuous linear functional $C_{0}(X)$ (= a signed bounded Radon measure) if and only if it is of bounded variation. Moreover, 2) and 3) coincide if $X$ is compact (and there are also proofs of 3) reducing it to that case).
