In situ Raman scattering is performed on an individual semiconducting double-walled carbon nanotube (DWNT) in a field-effect transistor (FET) geometry, while the transfer characteristics of the DWNT-FET are measured. Through studying the Raman spectra with response to forward and backward gate voltage (V(gs)) sweeping, respectively, we observe hysteresis loops in the curves of G(-) peak frequency and the intensity ratio of G(-) to G(+) (I(G(-))/I(G(+))) as a function of V(gs). These loops correlate very well with the hysteretic transfer characteristics of the device. The clear correlations suggest that G(-) peak line width and I(G(-))/I(G(+)) increase with the carrier concentration in the DWNT induced by V(gs). In addition, unique G(-) peak line width variations with V(gs) can be attributed to interband electron transitions between the energy bands of two concentric shells of the DWNT excited by G phonons.