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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?

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    Doesn't the standard proof via the Baire category theorem just work? – 2017-01-17
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    The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me. – 2017-01-17
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    HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range. – 2017-01-17

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Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X \rightarrow Y$ be continuous linear and $q: Y \rightarrow [0,\infty)$ be a continuous semi-norm. If $\sup \{q(T_a x):a\in A\} <\infty$ (pointwise bounded), then $\sup \{q(T_a x):a\in A\}$ is a continuous semi-norm (uniformly bounded/equicontinuous).

To prove this, let $$E_n = \{x\in X: q(T_a x) \le n, \forall a\}$$ By pointwise boundedness, we see that $E_n \nearrow X$. 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 $p_n$ and thus there exists $N,r$ such that $$ x + \bigcap_{k=1}^N \{p_k < r\} \subseteq E_n $$ Hence, $$ \bigcap_{k=1}^N \{p_k < 2r\} \subseteq E_n -E_n \subseteq E_{2n} $$ Let $$ p(x) = \frac{1}{r} \sum_{k=1}^N p_k (x) $$ Hence, $$ 0\in \{p < 1\} \subseteq \{\sup \{q(T_a x):a\in A\} <3n\} $$ 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.