Narrowband Signal Decomposition

Why do the in-phase and quadrature components of a narrowband signal each have half the bandwidth of the signal?

The in-phase and quadrature components of a bandpass signal are real-valued baseband signals. So if the spectrum of a bandpass signal with bandwidth $B$ extends from $f_c-B/2$ to $f_c+B/2$ (where $f_c>B/2$ is the carrier frequency), the spectra of the in-phase and quadrature components extend from $-B/2$ to $B/2$. Since bandwidth is measured at positive frequencies, the baseband bandwidth is $B/2$.