Same answer for both. It's an orthogonal transform, aka a change of basis. You're conceptually rotating the function/series to an equivalent one that's orthogonal to the original. The magnitude/energy hasn't changed (Parseval's theorem is a more succinct definition). And to perform the inverse transform you need to conceptually rotate it back to the original, which should mirror the original transform very nicely.
If it didn't preserve energy then there wouldn't necessarily be an inverse transform, since that implies information was lost.
If it didn't preserve energy then there wouldn't necessarily be an inverse transform, since that implies information was lost.