I learned that FT (Fourier Transform) of impulse train signal is impulse train.
I understood the intergral process, but I'm confused with the property of FT.
FT satisfied linearity, and impulse train can be expressed as sum of infite number of impulse signals at different times arranged at regular intervals.
Additionally, FT of impulse signal is 1 (over all frequencies).
Then why FT of impulse train is not infinite? Shouldn't it be sum of infinite 1s? like below
$\sum \delta(t-n) = \begin{cases} 1 & \text{if } t = n \\ 0 & \text{otherwise} \end{cases} = Ш(t)$
$\mathcal{F}(\sum \delta(t-n)) = \sum \mathcal{F}(\delta(t-n)) = \sum 1 = \infty$