# Is this estimate true for functionals on Frechet spaces?

In class the other day my professor made the following claim about the Schwartz class $$\mathcal S$$:

Let $$u: \mathcal S \to \mathbb C$$ be linear. $$u$$ is continuous iff there exist $$C,N>0$$ such that for all $$\varphi \in \mathcal S$$, $$|u(\varphi)| \leq C \sup_{|\alpha|,|\beta| \leq N} ||x^\alpha D^\beta \varphi||_{L^\infty}$$ where $$\alpha,\beta$$ denote multiindices.

(so $$||x^\alpha D^\beta \varphi||_{L^\infty}$$ is just the $$(\alpha,\beta)$$th seminorm of $$\mathcal S$$).

He claims this follows immediately from uniform boundedness for Frechet spaces. I'm largely unfamiliar with this area of functional analysis, so it isn't at all clear to me. In particular, I don't see why we'd only need to use finitely many seminorms.

So, is the claim

Let $$F$$ be a Frechet space with seminorms $$||\cdot||_j$$. A linear map $$u: F \to \mathbb C$$ is continuous iff there exist $$C,N>0$$ such that for all $$f \in F$$, $$|u(f)| \leq C\sup_{j \leq N} ||f||_j$$

valid, or is this just something true for the Schwartz class? Does it follow from Banach-Steinhaus?

The result is true in any locally convex space $$F$$. It is a consequence of the fact that the topology is given by a family of seminorms.
Since $$u$$ is continuous at $$0$$, the set $$u^{-1}(B_1(0))$$ is open (preimage of the unit disk). So there exists a neighbourhood $$V$$ of zero, of the form $$V=\{f:\ \|f(x)\|_j<\varepsilon,\ j=1,\ldots,N\}$$, with $$V\subset u^{-1}(B_1(0))$$. Given any $$f\in F$$, we have $$\frac{\varepsilon f}{2\sum_{j=1}^N\|f\|_j}\in V,$$ so $$\left|u\left(\frac{\varepsilon f}{2\sum_{j=1}^N\|f\|_j}\right)\right|\leq1,$$ which we may write as $$|u(f)|\leq \sum_{j=1}^N\frac2\varepsilon\,\|f\|_j\leq\frac{2N}\varepsilon\,\sup_{j\leq N}\|f\|_j.$$
There is a small exception to the above, which is the case in which $$\|f\|_j=0$$ for $$j=1,\ldots,N$$. But in that case $$mf\in V$$ for all $$m\in\mathbb N$$, so $$|u(mf)|\leq1$$ and so $$|u(f)|\leq1/m$$, forcing $$u(f)=0$$.