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!

dualis usually associated to a vector space. The space of probability measures is not a vector space. – Liviu Nicolaescu Dec 5 '13 at 21:28