When can one pass a linear operator under the integral?

Question

It is written that $$ \int\mathcal{L}G(x,s)f(s)ds = \mathcal{L}\left(\int G(x,s)f(s)ds\right) $$

where $G$ is a Green function. Why can the linear operator be pulled out of the integral?

Thanks!

score 0 · Answer 1 · answered Sep 14 '17 at 19:50

0

Because when $x$ is fixed the integral is a limit of linear combinations of $G(x,s_i)$'s.

answered Sep 14 '17 at 19:50

uniquesolution

3

I think this answer is pretty misleading; one of the main issues that makes analysis hard is that limits and integrals frequently do not commute. – Sep 14 '17 at 19:53
@user296602 You are right, or course, but clearly the necessary condition I have indicated is also sufficient in this case, by virtue of various assumptions that the OP has neglected to mention. – uniquesolution Sep 14 '17 at 19:55
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Could you please elaborate on your answer? – Quoka Sep 14 '17 at 20:25
@Quoka the problem with the answer is the interchanging limit with linear operators, which is only true if the operator is continuous. – quallenjäger Feb 12 '19 at 14:49

1 Answers1