The Banach-Steinhaus theorem for seminormed spaces Assume that we have a vector space $X$ over reals  with a countable sequence of  seminorms $p_n$ on $X$ such that:
$$
p_n(x)\leq p_{n+1}(x) \textrm{ for } n\in \mathbb N, x\in X, 
$$
$$
\textrm{ for } x\in X\setminus \{0\} \textrm{ there is } n\in \mathbb N \textrm{ such that } p_n(x)\neq 0. 
$$
Then $X$ is a metric space with the metric
$$
d(x,y)=\sum_{n=1}^\infty \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}, \  x,y\in X.
$$
Such a space $X$  is called the countably seminormed space.
Let's consider in a complete countably seminormed space $X$ a sequence
$T_k:X\rightarrow \mathbb R$ of linear continuous mappings such that for each $x\in X$ the sequence of numbers $(T_k(x))_{k \in \mathbb N}$ is bounded. Then by the Banach-Steinhaus theorem for the Frechet spaces  the family $(T_k)_{k\in \mathbb N}$ is equi-continuous. Moreover, by properties of linear functionals on countably seminormed spaces, for each $k\in \mathbb N$ there exists $N_k$ such that $T_k$ is continuous with respect to the seminorm $p_{N_k}$. I.e. $|T_k(x)|\leq M p_{N_k}(x)$ for all $x\in X$, where $M>0$ is some constant depending on $k$.
Does there then exist an $N\in \mathbb N$ and $M>0$, not depending on $k\in \mathbb N$, such that
$$
|T_k(x)|\leq M p_N(x) \textrm{ for } x\in X, k \in \mathbb N ?
$$

Yes, this is equicontinuous. Since the topology is defined by a sequence of seminorms, so you can define the equicontinuity by the $p_k$. 
This is the problem about Banach-Steinhaus  theorem on Frechet space (or metric linear spaces.)
Maybe you can see Rolewicz'book "Metric Linear Spaces" or goolge for more about this problem.
