Sequential convergence of the compact-open topology, where the codomain has weak topology. Let $X$ be a topological space, $Y$ be a locally convex space equipped with the weak topology. Let $C(X, Y)$ denote the set of all continuous maps between $X$ and $Y$. Now endow $C(X, Y)$ the compact-open topology. My question is:

Is there any characterization for the sequential convergence on
  $C(X, Y)$, like $f_n\to f$ in $C(X, Y)$ if and only if $\langle f_n,\phi\rangle \to \langle f,\phi\rangle$ in $C(X, \mathbf R)$ for all $\phi\in Y^*$?

Here $Y^*$ denotes the topological dual of $Y$. But I have no idea to prove or disprove that. The key difficulty is that the weak topology on the codomain $Y$ is not metrizable in general. Can anyone give some hints or reference? TIA...
 A: For generality, we only suppose the space $Y$ to be originally a locally convex space with the topology induced by a family $\mathcal P$ of seminorms. In this case, $Y$ is also a uniform space, with the uniform structure defined by the family of pseudometrics associated with the seminorms in $\mathcal P$. Hence, the compact-open topology on $C(X,Y)$ is equal to the topology of compact convergence.
For a net $\{f_\alpha\}_{\alpha\in A}\subset C(X,Y)$ and $f\in C(X,Y)$, $f_\alpha\to f$ is equivalent to $f_\alpha|_K\to f|_K$ uniformly for all compact subset $K\subset X$. Besides, the uniform convergence $f_\alpha|_K\to f|_K$ is equivalent to for every entourage $V$ in $Y\times Y$, there exists an $\alpha_0$, such that for every $x\in K$ and every $\alpha\ge \alpha_0$, $(f_\alpha(x),f(x))\in V$. Moreover, the following subsets of $Y\times Y$
\begin{split}
V & = \Big(\max_{i=1,\cdots, n} p_i\Big)^{-1}([0,\epsilon]) \\
& = \{(y_1,y_2)\in Y\times Y: p_i(y_1-y_2) \le \epsilon, \forall i=1,\cdots, n\}, \quad p_i\in\mathcal P, n\in\mathbf N_+, \epsilon >0.
\end{split}
form a fundamental system of entourages. Hence, $f_\alpha|_K\to f|_K$ is also equivalent to for every $p\in\mathcal P$ and $\epsilon>0$, there exists an $\alpha_0$, such that for every $x\in K$ and every $\alpha\ge \alpha_0$, $p(f_\alpha(x)-f(x)) \le \epsilon$. We thus proved the following theorem.

Theorem. Let $X$ be a topological space, $Y$ be a locally convex space with the topology induced by the family $\mathcal P$ of seminorms. Let $C(X,Y)$, $C(X,\mathbf R)$ denote the space of all continuous maps between $X$ and $Y$, $X$ and $\mathbf R$, respectively, both endowed with the compact-open topology. Then a net $\{f_\alpha\}$ in $C(X,Y)$ converges to $f$ if and only if for every $p\in\mathcal P$, $p(f_\alpha-f)\to0$ in $C(X,\mathbf R)$.

In particular, if the locally convex space $Y$ is equipped with the weak topology, then the family $\mathcal P = \{p_\phi: p_\phi(y) = |\langle y, \phi\rangle|, y\in Y,\phi\in Y^*\}$. Consequently,

Lemma. With the assumptions in the previous theorem, if $Y$ is now equipped with the weak topology (instead of the original topology induced by $\mathcal P$), then $f_\alpha\to f$ in $C(X,Y)$ if and only if for every $\phi\in Y^*$, $\langle f_\alpha,\phi\rangle \to \langle f,\phi\rangle$ in $C(X,\mathbf R)$.
Corollary. With the assumptions in the previous lemma, for every $\phi\in Y^*$, the mapping
\begin{equation}
\begin{split}
\phi^*: C(X,Y) &\to C(X,\mathbf R), \\
f &\mapsto \langle f,\phi\rangle,
\end{split}
\end{equation}
is continuous.

Here the star on the mapping $\phi^*$ indicates that it is the pushforward of $\phi$.

Some remarks for the set $C(X,Y)$.
We will indicate the topology on $Y$ in the following way: by $(Y,\mathcal P)$ we mean the topology of $Y$ is the original locally convex topology induced by the family $\mathcal P$ of seminorms, by $(Y,\text{wk})$ we mean $Y$ is equipped with the weak topology.

*

*The locally convex topology on $Y$ induced by the family $\mathcal P$ of seminorms is nothing but the initial topology induced by $\mathcal P$. Namely, it is the coarsest topology for which all the mappings $p(\cdot-y), p\in\mathcal P,y\in Y$ are continuous. By the characteristic property of the initial topology, $f\in C(X,(Y,\mathcal P))$ if and only if $p(f(\cdot)-y)$ is continuous for every $p\in\mathcal P$ and $y\in Y$, if and only if $p\circ f\in C(X,\mathbf R)$ for every $p\in\mathcal P$, since $f\in C(X,(Y,\mathcal P))$ is equivalent to $f(\cdot)+y\in C(X,(Y,\mathcal P))$, for any $y\in Y$. In particular, if $Y$ is equipped with the weak topology, then $f\in C(X,(Y,\text{wk}))$ if and only if $|\langle f,\phi\rangle|\in C(X,\mathbf R)$ for all $\phi\in Y^*$.


*We can use the net convergence to get a stronger version. Indeed, $f\in C(X,(Y,\mathcal P))$ if and only if for all net $\{x_i\}$ in $X$ converging to $x$, the net $\{f(x_i)\}$ converges to $f(x)$, if and only if
$$(x_i\to x) \Rightarrow (\forall p\in\mathcal P, p(f(x_i)-f(x))\to0).$$
If $Y$ is equipped with the weak topology, then $f\in C(X,(Y,\text{wk}))$ is equivalent to
$$(x_i\to x) \Rightarrow (\forall \phi\in Y^*, \langle f(x_i), \phi\rangle \to \langle f(x), \phi\rangle ),$$
that is, $\langle f,\phi\rangle\in C(X,\mathbf R)$ for all $\phi\in Y^*$.


*If moreover, the original locally convex topology on $Y$ is completely normable so that it admit a norm $\|\cdot\|$ to be a Banach space, and $f$ satisfies $\sup_{x\in X}\|f(x)\| <\infty$, then by the uniform boundedness principle, $f\in C(X,(Y,\text{wk}))$ if and only if $\langle f,\phi\rangle\in C(X,\mathbf R)$ for all $\phi\in M$ with $M$ a fixed dense subset of $Y^*$.


*Of course, one can easily apply the above arguments to the case of $C(X,Y^*)$ with $Y^*$ equipped by the weak-$*$ topology.


*In all the above arguments, the most important stuff is the uniform structure on $Y$, the algebraic structure is not yet necessary. Indeed, if $Y$ is only a uniform space, on which the uniform structure is generated by a family of pseudometrics $\mathcal D$ (called gauge in John L. Kelley's book), then $f\in C(X,(Y,\mathcal D))$ if and only if for all net $\{x_i\}$ in $X$ converging to $x$, the net $\{f(x_i)\}$ converges to $f(x)$, if and only if
$$(x_i\to x) \Rightarrow (\forall d\in\mathcal D, d(f(x_i),f(x))\to0).$$
A net $\{f_\alpha\}$ in $C(X,(Y,\mathcal D))$ converges to $f$ if and only if for every $d\in\mathcal D$, $d(f_\alpha,f)\to0$ in $C(X,\mathbf R)$.
