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 $\tau_2$).

  • $\begingroup$ Ok, where can I read about that if I would like to study which topologies are needed for a given $T$. Thanks $\endgroup$ – HFKy Oct 12 '18 at 12:33
  • $\begingroup$ 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". $\endgroup$ – ald.li Oct 12 '18 at 15:01
  • $\begingroup$ Great!! Thanks for the help!! $\endgroup$ – HFKy Oct 12 '18 at 15:25

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