Continuity of an operator on a Fréchet space

[slender pine]

Consider a Fréchet space $F$ endowed with a countable family $N_p$, $p \in \mathbb{N}$ of seminorms.
I'm trying to figure out what continuity means for a linear operator on $F$. I know that a sequence $(\phi_n)$ of a is said to converge to $\phi$ when $N_p(\phi_n - \phi)$ for all $p$. Now, consider $A : F \to F$ a linear operator. As far as I read on books/internet, I'm thinking of two possible definitions.\
The first : $A$ is continuous on $F$ iff there exist $C > 0$ such that $N_p(A\phi) \leq C N_p(\phi)$ for all $\phi \in F$.\
The second : $A$ is continuous on $F$ iff there exist a strictly increasing function $\varphi : \mathbb{N} \to \mathbb{N}$ and a constant $C > 0$ such that $N_p(A\phi) \leq C N_{\varphi(p)}(\phi)$ for all $\phi \in F$.\
In the first definition, we ask to control each seminorm of $A\phi$ by the same seminorm of $\phi$. In the second, we just ask every seminorm of $A\phi$ to be controlled by any finite number of seminorms of $\phi$. I can't see which one is correct (I'm leaning towards the 2d but I'm not sure)

[hardy lake]

Twenty

[south cove]

what version of math is this!!??

[real crow]

Try asking in #advanced-analysis

You'll get better responses

[slender pine]

Okay, thanks

.solved