What are the mathematical consequences of a discrete signal living in Hilbert space?
whether or not a signal lives in a Hilbert space depends only on if you say it does.  if you say so, then you need to specify an inner product $&lt;x,y&gt;$ (which is pretty easy for finite-length discrete-time signals)

$$ \langle x,y \rangle = \sum_n x[n] y^*[n] $$

and you need to use that inner product definition to define a norm

$$ |x| = \sqrt{\langle x,x \rangle} $$

which should also be the same as the distance measure from $x$ to the $0$ element in that Hilbert space.

and you need to be able to add and subtract elements in the Hilbert space (easy to do for discrete-time signals).  why this might be useful would be for defining a signal $x$, an estimation of that signal $\hat{x}$, and an error signal, $e_x$ such that

$$ \hat{x} = x + e_x $$
or
$$ e_x = \hat{x} - x $$

we can think of the norm of $e_x$ which is

$$ |e_x| = \sqrt{\langle e_x,e_x \rangle} $$

as a measure of the distance that $\hat{x}$ is from $x$.

if gives you a formal framework for defining error and then maybe help you with algorithms (like a matched filter) for minimizing error, so that a received signal most closely approximates what was transmitted.  or so that some other  "thing" most closely hits some specified target "thing".