$EV(x): x \mapsto ev_x$ is continuous Let $M$ be Locally compact Hausdorff and $C_0(M)$ denote the space of real valued continuous functions that vanish at infinity.  My questions reads: ''Show that $EV$ is a homeomorphism onto its image $EV(X)$ equipped with the weak* topology.''
To show continuity I tried;
Let $(x_n)$ be a sequence in $M$ converging to $x$, fix $f \in C_0(M)$. Calculate;
$$\lim_{n\rightarrow \infty}EV(x_n)f=\lim_{n \rightarrow \infty}ev_{x_n}f=\lim_{n\rightarrow \infty}f(x_n)=f(x)=EV(x)f$$
But this dosn't make any sense to me. I never touched the topology on $C_0(M)^*$, which would imply that $EV$ is always continuous no matter what open sets look like in $C_0(M)^*$? But isn't the whole point of the weak star topology to force the maps $ev_x$ to be continuous. What am I doing wrong?
 A: You say that you did not touch the topology of $C_0(M)^*$. You did!
Let's recall. Assume that $X$ is a topological vector space. Then we define the topological dual $X^*$ as $\{\varphi:X\to\mathbb{K}\vert\;\;  \varphi \text{ is continuous on }X\}$.
and we endow $X^*$ with the weak-star topology; this is a locally convex topology on $X^*$ that is produced by the family of seminorms $\{p_x\}_{x\in X}$, where $p_x(\varphi)=|\varphi(x)|$. It is easily verified that a net $\{\varphi_i\}\subset X^*$ converges to $\varphi\in X^*$ with respect to the weak-star topology if and only if for all $x\in X$ we have that $\varphi_i(x)\to\varphi(x)$ (and this happens of course in $\mathbb{K}$ and its standard topology).
Now you have a map $\text{ev}:M\to C_0(M)^*$ (where $M$ has its topology and $C_0(M)^*$ has the weak star topology coming from being the dual of $C_0(M)$) and you are showing that $\text{ev}$ is continuous with respect to the topologies just mentioned. So, you need to show that, if $\{x_i\}\subset M$ is a net in $M$ converging to $x\in M$, then $\text{ev}(x_i)$ converges to $\text{ev}(x)$ in $C_0(M)^*$ in the weak-star topology. As we explained, we have to show that for every element $f\in C_0(M)$, we have that $\text{ev}(x_i)(f)\xrightarrow[i\in I]{}\text{ev}(x)(f)$ (this happens in $\mathbb{K}$). This is precisely what you did, using of course the fact that when $f\in C_0(M)$, then $f$ is continuous as a function $f:M\to\mathbb{K}$.
