# Infinite lineal combination on a space of functions, which convergence do I need?

I have been wondering this question for a long time without getting an answer, I just arrive to necessary conditions but I would like to get a characterization of the problem. The problem is, if I have a linear operator, $$T:V \to V$$, on a space of functions $$V$$ then,

When is true that $$T(\sum^{\infty}\alpha_nf_n) = \sum^{\infty} \alpha_nT(f_n)?$$

In order to get a necessary conditions, if $$V = C^{\infty}(\mathbb{R})$$ and $$T$$ maps $$f \mapsto f'$$, then I need that there exist $$x_0$$ such that $$\sum^{\infty}\alpha_nf_n(x_0)$$ converges and also that $$\sum^{\infty}\alpha_nf'_n$$ converges uniformly on $$\mathbb{R}$$. To sum up, I need $$C^1-$$converge.

Indeed, one can extend the previous argument to see that it is needed $$C^{\infty}-$$convergence, for example take $$T$$ to be the second derivative, then we need $$C^2-$$convergence, and so do.

I would appreciate if someone could shed some light on the matter. Maybe we need to discuss about conditions on both, $$T$$ and $$V$$.

You are just talking about continuous operators in different linear topologies. Your condition is equivalent to $$T$$ being continuous from $$(V,\tau_1)$$ to $$(V, \tau_2)$$ for some linear topologies $$\tau_1, \tau_2$$ (the first infinite sum is taken with respect to $$\tau_1$$, the second with respect to $$\tau_2$$).
• Ok, where can I read about that if I would like to study which topologies are needed for a given $T$. Thanks – HFKy Oct 12 '18 at 12:33
• The main point is that for linear operators your infinite sum condition is equivalent to just being (sequentially) continuous (i.e. $T(\lim x_n) = \lim Tx_n$ for all converging sequences $(x_n)$). For studying continuity it is enough to brush some introductory chapters on Linear Topology (e.g., Aliprantis-Burkinshaw "Hitchhiker's guide to Infinite-dimensional Analysis" is fine). Concerning specifically derivation operators on function spaces, I guess you want to go with Operator Theory introductions. Consider, e.g., Davies' "Linear Operators..." or "Spectral theory and Diff. operators". – ald.li Oct 12 '18 at 15:01