0
$\begingroup$

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 $V$.

$\endgroup$
1
$\begingroup$

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 $\tau_2$).

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.