In Wikipedia (link), we have the following definition:
$S$ is a metric space, considered with its Borel $\sigma$-algebra. Then, a sequence of Borel probability measures $P_n$ on $S$ converges weakly to a Borel probability measure $P$ on $S$ if for every bounded continuous function $f_n:S\rightarrow\mathbb{R}$, $\int fdP_n\longrightarrow\int fdP$.
It is noted that this is "actually" weak-* convergence.
How is it weak-* convergence? For weak-* topology, I need a normed space, and then I need to take a dual and there I have weak-* convergence. Probably the space of probability measures lives as a subspace of this dual.
I am confused about this, and would appreciate some accurate definitions of weak and weak-*, and a description of the relations between them.
I only care about the Borel probability measures on $S$, and I mostly care about the case where $S$ is a compact metric space.