Let $E$ be an LF-space (see page 126 of $[1]$), $F$ a locally convex space and $f: E \to F$ a linear map. I want to prove that $f$ continuous if and only if it is sequentially continuous.
I tried the following: let $(E_n)_{n \in \mathbb{N}}$ be a sequence of definition of $E$.
$\Rightarrow)$ Since $f$ is continuous, then $f{_{|_{E_n}}}$ is continous too, for all $n \in \mathbb{N}$, with $f$ linear and $F$ locally convex. Thus, $f{_{|_{E_n}}}$ is sequentially continuous. But, how to conclude that $ f $ is sequentially continuous? Here $f{_{|_{E_n}}}$ means that $f:E_n \to F$, for each $n \in \mathbb{N}$.
$\Leftarrow)$ Since $f$ is sequentially continuous, then $f{_{|_{E_n}}}$ sequentially continuous, for all $n \in \mathbb{N}$ (it's true?). Therefore, $f{_{|_{E_n}}}$ is continuous and hence $f$ is continuous.
The implication $\Leftarrow)$ it's right? and in $\Rightarrow)$ how to conclude that $f$ is sequentially continuous?
Previously I used the heavily Propositions $13.1$ and $8.5$ [$1$].
[1] Trèves, F. Topological vector spaces, distributions and kernels. Unabridged republication of the $1967$ original. Dover Publications, Inc., Mineola, NY, $2006$.
$(\Rightarrow)$ Assume that $f$ is continuous. Given $x_n \rightarrow x$, we wish to show that $f(x_n) \rightarrow f(x)$. Let $V$ be a neighborhood of $f(x)$. Then $f^{-1}(V)$ is a neighborhood of $x$, and so there is an $N$ such that $x_n \in f^{-1}(V)$ for $n \geq N$. Then $f(x_n) \in V$ for $n \geq N$.
$(\Leftarrow)$Suppose that $f:E\rightarrow F$ is sequentially continuous. We claim that $f|_{E_j}:E_j \rightarrow F$ is sequentially continuous for all $j \in \mathbb{N}$. Indeed, let $x_n \rightarrow x$ in $E_j$, since $i_j:(E_j,\tau_j) \rightarrow (E,\tau_{ind})$ is continuous, then $x_n \rightarrow x$ in $E$. Therefore, $f_{E_j}(x_n)=f(x_n) \rightarrow f(x)=f_{E_j}(x)$ in $F$. From Proposition 8.5 in Trèves book it follows that $f|_{E_j}$ is continuous for all $j \in \mathbb{N}$. Taking into account Proposition 13.1 it follows that $f:E \rightarrow F$ is continous.