Let $(\mathcal{O},\tau_{ind})$ denote the space of germs of all $\mathbb{C}$-valued-holomorphic functions at 0 equipped with the inductive topology which turns it into a Silva/DFS-space (i.e. it is the inductive limit of a sequence of Banach spaces, with compact linking maps). Another description of $\mathcal{O}$ is that it is the ring of all power series which converge locally at 0. $(\mathcal{O},\tau_{ind})$ is isomorphic (as LCVS) to the strong dual of $(\mathcal{H}(\mathbb{C}),\tau_{co})$ - the space of entire functions equipped with the compact-open topology, which can be also described as a power series space (see for example the introductory book on functional analysis by Meise and Vogt). The duality is $ := \sum a_{n} \phi_{n}$. For $\phi = \sum \phi_{n} x^{n} \in \mathcal{H}(\mathbb{C})$ let $p_{\phi}(\sum_{n \in N} a_{n} x^{n}):= \sum \vert \phi_{n} a_{n} \vert$. The system $\{p_{\phi} \mid \phi \in \mathcal{H}(\mathbb{C})\}$ then generates the inductive topology on $\mathcal{O}$. My question: Does $(\mathcal{O},\tau_{ind})$ carry the weak topology? I.e. is it true that a net $(f_{i})_{i\in I}$ converges to 0 in $(\mathcal{O},\tau_{co})$ iff $
Weak topology on the space of germs of holomorphic funtions
0
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complex-analysis
functional-analysis
1 Answers
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The simple answer is---no. One way to see this is to use the fact that this space with the weak topology is not complete but every Silva space is. The only situation where a result of this kind can hold is the special one of a Silva space which is the inductive limit of finite dimensional spaces, essentially the space of finite sequences.
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0In fact, the strong dual of Frechet-space $E$ never carries the weak topology, unless $E$ is finite-dimensional (see the article "When $(E,\sigma (E, E')) $ is a $ DF $-space?" by D. Krassowska. – 2012-11-09