Uniform Boundedness principle for bounded linear maps from Frechet Space into a Banach Space I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space. 
This is used in proving the space of tempered distributions is complete but I can't find a proof of it anywhere.
When I try to prove it myself I get stuck on the final part(which uses the scaling property of linear maps).
Does anyone have a proof that they could share? 

Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X \rightarrow Y$ be a continuous linear map for each $a\in A$, and let $q: Y \rightarrow [0,\infty)$ be a continuous semi-norm.
If $\sup \{q(T_a x):a\in A\} <\infty$ $\forall x\in X$ (pointwise bounded), then $x\mapsto \sup \{q(T_a x):a\in A\}$ is a continuous semi-norm (i.e., the set $\{T_a : a\in A\}$ is uniformly equicontinuous/uniformly bounded).
To prove this, let 
$$E_n = \{x\in X: q(T_a x) \le n, \forall a\}$$
By pointwise boundedness, we see that $\cup_n E_n = X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_k$ and thus there exists $N,r$ such that
$$
x + \bigcap_{k=1}^N \{p_k < r\} \subseteq E_n
$$
Hence,
$$
\bigcap_{k=1}^N \{p_k < 2r\} \subseteq E_n -E_n \subseteq E_{2n},
$$
because if $a\in \bigcap_{k=1}^N \{p_k < 2r\} $, then $a=(x+a/2)-(x-a/2)\in E_n-E_n$.
Let
$$
p(x) = \frac{1}{r} \sum_{k=1}^N p_k (x).
$$
Hence,
$$
0\in \{p < 1\} \subseteq E_{2n}\subseteq \{x: \sup \{q(T_a x):a\in A\} <3n\}.
$$
Since $p$ is continuous, we see that $0$ is an interior point of $\{\sup \{q(T_a x):a\in A\} <3n\} $ and thus $\sup \{q(T_a x):a\in A\}$ is a continuous semi-norm.
