TITLE: Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?
QUESTION [7 upvotes]: Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am requesting help here.
The question I face is:

If we have a domain (like the unit disk) D in the complex plane and we have the topological algebra of analytic functions in one variable on D, is there a suitable norm on this algebra such that it induces the "natural topology" i.e. the topology of convergence on compact sets?

I would truly appreciate a help in the form of a yes/no response with some convincing to settle this for me. I thank all kind helpers.

REPLY [8 votes]: By Montel's theorem, every bounded set (w.r.t. the topology of uniform convergence on compact sets) in the space of holomorphic functions is relatively compact. If the space were normed its closed unit ball would be compact which implies that the space is finite dimensional.