TITLE: dual space of the subspace of the space of probability measures
QUESTION [5 upvotes]: I have a question which maybe so naive but I want to know the result about it.
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as Multidimensional diffusion processes and Large deviations, we know that the dual space of $\mathcal{M}$ is $\mathcal{C}_b^0(\mathbb{R})$ which is the space of continuous bounded functions defined on $\mathbb{R}$. Here the topology of $\mathcal{M}$ is induced by weak convergence.
Now we consider a subspace $\mathcal{M}_p$ of $\mathcal{M}$ such that:
$$\mathcal{M}_p=\{\mu\in\mathcal{M}: \int_{\mathbb{R}} x^2\mu(dx)<\infty\}$$
I would like to know the dual space of $\mathcal{M}_p$, I guess it is the space of continuous functions $f$ satisfying 
$$|f(x)|\leq C(1+|x|^2)$$
for some constant $C$. But I don't know how to prove it. If someone knows it please let me know. Thanks a lot!

REPLY [3 votes]: If $X$ is any locally convex space and $L$ is a subspace (endowed with the relative topology) then the (continuous) dual $L'$ of $L$ is a quotient of $X'$ by the subspace $L^\perp=\lbrace f\in X': f|_L=0\rbrace$
(this follows from Hahn-Banach: the restriction map $X' \to L'$ is surjective).
Even if you modify the definition of $\mathcal M_p$ as suggested by Gerald Edgar
there is thus no reason to expect $\mathcal M_p'$ to be a  subspace of $\mathcal M'$.