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Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$

Is there a common topology to place on this space, if yes what is the topology and is it induced by a norm?

What can be said about differential operators over this space? Are they bounded/unbounded? Would there be a natural topology to place on the space of differential operators over $\mathcal{A}(-a,a)$?

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