Linearity with Difference Period in Fourier Transform I know that a system is linear if it satisfies

$$\mathscr{F}\{ a\,x(t)+b\,y(t) \} = a\,X(\omega)+b\,Y(\omega)$$ 

for Fourier transform, $X(\omega)\triangleq\mathscr{F}\{x(t)\}$

But what if $x(t)$ and $y(t)$ don't have the same period? 

Can we use same linearity condition in Fourier transform?
Continuous time Fourier transform does not require the signals to be periodic, and therefore the linearity assumption holds. CTFT is a linear operator.

Furthermore, if you mean CT Fourier series, which require that both signals to be periodic by (a common period) $T_0$, that's also linear.

But if input signals have no common periods, then you cannot apply the linearity test for CTFS.