Is it customary to correct for the gain of a window? Consider how the Hanning window is defined:

0.5 - 0.5 * cos(n*2*Pi/(N-1))


By this definition, it has a gain of 0.5, which is simply the average value of the coefficients. By contrast, Flattop windows, as defined, have unity gain, presumably by design. 

It would seem appropriate to scale the Hanning window by a factor of 2, but I have never seen this discussed anywhere. It would seem that all windows should be scaled for unity gain. 

In practice, are windows typically corrected for their gain?  If not, why not?

EDIT:

Since nobody has given an answer, I'll elaborate a bit.

It is quite easy to find papers that report the gain of the more common windows. But nowhere have I seen anyone refer to correcting the gain before using it for spectral analysis. Maybe I have always missed that statement, or everyone assumes gain correction to be an obvious requirement.

It seems like common sense to set the gain of a window to unity so that signal's energy level is preserved. Furthermore, how can one compare the various windows for amplitude accuracy if one has 0 dB gain, as a flattop does, and the other has nearly 10 dB loss, as the Gauss does.

Windows are also widely used for FIR filter design. In this application, it should be clear that the signal to be windowed, a sinc pulse, has most of its energy in the center of the window. Consequently, the window does little to reduce the sinc pulse's total energy. Thus, when used for filter design, we don't want unity gain, but rather unity peak amplitude, as most windows have, except the flattops. Something other than unity peak amplitude would affect the gain of the resulting FIR filter.
Yes, it is customary to correct for the gain of a window, except for some cases I refer to later. (If you are interested only in the relative amplitude, of course you do not need to correct for the gain.) 

Because the window reduces the gain of the original signal (time domain), the amplitude obtained through FFT need to be corrected. For example, if you use the Hanning window, you need to multiply all the amplitudes by 2 (the reciprocal of 0.5). As I understand it, most of the software packages for FFT automatically correct for the window used.

However, such correction is good only when all the frequencies of interest distribute throughout the time domain window. For example, suppose you have 1024 data with all signal levels of zero except for #512 point which has a value of 1 (impulse signal). Obviously, any windows do nothing to the data. So, if you correct the amplitudes for the window gain (multiply by 2), then you will end up with overestimation of the amplitude. If your 1024 data are all zero except for the very 1st point with a value of 1, then every point has a value of zero after windowing, and you lose the signal.

So, if you are dealing with random signals, with all the frequency components expected to lie almost evenly over the length of the signal, you need (or should) correct for the gain of the window you use.