When the conditions of the sampling theorem are not fulfilled we have aliasing. The distortion introduced by aliasing cannot be undone in the discrete-time domain, so aliasing must be avoided before sampling. This is done by first lowpass filtering the signal with a filter with cutoff frequency $$F_s/2$$. This operation can be interpreted within the setup of the Hilbert space we have studied previously. This initial filtering operation is an orthogonal projection of the signal onto the subspace of bandlimited signals, from which it can be perfectly sampled and reconstructed at sampling frequency $$1/T_s$$

Sometimes, when the signal is bandpass, alias-free sampling can be achieved at rates below the nominal rate stipulated by the sampling theorem. This operation is called bandpass sampling and it's a useful exercise to understand what happens when we move from the continuous-time to the discrete-time domain.