The function $$\mbox{sinc}(t/T_s)$$, whose zeros fall at multiples of $$T_s$$, is $$F_s$$-bandlimited, where $$F_s = 1/T_s$$. When we interpolate a sequence with this sinc interpolator, we obtain a continuous-time signal $$x(t)$$

$$x(t) = \sum_{n=-\infty}^{\infty} x[n] \mbox{sinc}\left( \frac{t - nT_s}{T_s} \right)$$

whose continuous-time Fourier transform is a scaled version of the DTFT of $$x[n]$$ on the interval $$[-F_s/2, F_s/2]$$ and 0 otherwise. The interpolated function is thus bandlimited.

If we take all $$F_s$$-bandlimited functions, we obtain a Hilbert space and the set of rescaled and shifted sinc function forms an orthonormal basis for this space. In other words, we can interpret sampling as the basis expansion

$$x[n] = \langle \mbox{sinc} \left( \frac{t - nT_s}{T_s} \right), x(t) \rangle = T_s x(nT_s)$$

and the interpolation as a reconstruction formula

$$x(t) = \frac{1}{T_s} \sum_{n=-\infty}^{\infty} x[n] \mbox{sinc}\left( \frac{t - nT_s}{T_s} \right)$$

Therefore any $$F_s$$-banldimited function $$x(t)$$ can be represented with no loss of information by a sequence of samples taken every $$T_s$$ seconds, as long as $$T_s \leq 1/F_s$$.