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I have a doubt. When we have a Banach space $X$, by definition, each element $f$ in $X$ has a norm $\left\|f\right\|_X$. On the other hand, I understand that the space of entire functions $\mathcal{O}(\mathbb{C}):=H$ is not a Banach space but if it is a Frechet space and the topology obtained is the one obtained from the norms $\left\|f\right\|_K:=sup_{z\in K}|f(z)|$ with $K$ a compact subset of $\mathbb{C}$. Is it possible to define $\left\|f\right\|_H$ in some way? I mean if $\left\|f\right\|_H=sup_{z\in K}|f(z)|$ for some $K$ compact subset of $\mathbb{C}$? (I can't understand this concept) Thank you.

eraldcoil
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  • I would like to prove that the operator $M_z$ multiplication by $z$, $M_z:H\to H$ is bounded, that is, that $\left|M_z f(z)\right|{H}=\left|zf(z)\right|{H}\leq C \left|f\right|_H$

    The same for the differentiation operator $\partial_z$, which is bounded. My goal is to try to understand what it means for an operator to be limited between these spaces with those rules.

     – eraldcoil Sep 27 '23 at 03:08

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