I'm taking a signal processing class and I've become quite confused about the nature of the subject. We began the course by elaborately developing the theory of function spaces (infinite dimensional Hilbert spaces), but only after this did I realize that signals are finite discrete functions. So it's always necessarily to embed these finite discrete-time signals in the function space in ways that seem far-fetched and rather pointless.
Take the DFT for example. We have a signal $x[n]$ of finite length, $0\leq n < N-1$ and we embed this in $L^2([0,2 \pi])$ by interpreting the components as sample values for some periodic function on $[0,2\pi]$. Then we apply the Fourier transform to get the coefficients of the Fourier series. This eventually leads to defining the space of $P_N$ of N-periodic functions with basis of the sampled complex exponentials.
The story is similar for the Haar wavelet basis. The decomposition is defined on the entire continuous space $L^2(\mathbb{R})$, and after all the effort of embedding we get a discrete transform. My question is: why not just define the discrete transforms independently, without this continuous Hilbert space machinery?
Have you tried discussing this with your instructor? What books have you tried reading? Some standard textbooks you might appreciate are Oppenheim and Schafer, Haykin, and Mallat. Mallat's book is the most mathematical and you will notice much of it is "continuous."
Be careful not to overestimate the distinction between continuous and discrete signals. Signal processing is at least partly an engineering discipline. 20th century technology being what it is, that means we often want to actually compute something --- and computing things often involves approximating infinite things with finite ones (e.g. finite sums approximating Taylor series, finite difference equations approximating differential ones).
What you seem to miss is a lot of the signals we want to analyze are fundamentally infinite dimensional. My speech isn't really a finite length, discrete-time signal $\{x[n] \, \mid \, 1 \leq n \leq N\}$, right? If you put a microphone in front of me, then it's getting buffeted by a sound wave that has some intensity $\{x(t) \, \mid \, t \in [0,T]\}$. That's what we're really interested in.
Let's say the microphone records $N$ instances of $\{x(t) \, \mid \, t \in [0,T]\}$, specifically $\{x(iT/N) \, \mid \, 0 \leq i \leq N - 1\}$. We're losing sight of our goal (understanding $t \mapsto x(t)$) if we start working in $\{0,1,\dots,N-1\}$ --- hence it makes sense to embed $\{x(iT/N) \, \mid \, 0 \leq i \leq N - 1\}$ in $[0,T]$ via something like $\hat{x}(t) = \sum_{i = 0}^{N-1} x(iT/N) 1_{[iT/N,(i+1)T/N]}$. This is especially true since we might be interested in more than one possible choice of $N$ or other interpolants (more on this in the last paragraph).
To appreciate why Hilbert spaces come into play, my understanding is something like this: say your phone wants to send some bits $\{a_{1},\dots,a_{N}\} \subseteq \{-1,1\}^{N}$ to my phone. Let's say the way it is does this is its antenna emits an electromagnetic signal $x^{a}$ given by \begin{equation*} x^{a}(t) = \sum_{i = 1}^{N} a_{i} \cos(2 \pi k_{i} t) \end{equation*} with $k_{1},\dots,k_{N} \subseteq \mathbb{Z}$ some distinct set of frequencies. How does my phone figure out what $\{a_{1},\dots,a_{N}\}$ is? The matched filter works in the following way: for each $b \in \{0,1\}^{N}$, I have a "filter" that takes $x^{a} \to \langle x^{a},x^{b} \rangle$, where $\langle x^{a},x^{b} \rangle = \int_{0}^{2 \pi} x^{a}(t) x^{b}(t) \, dt$. What good is this? Well, you can check (by Cauchy-Schwarz) that $\langle x^{a},x^{b} \rangle \leq N$ with equality if and only if $a = b$. So if my filter computes each of these inner products, then I can figure out what $a$ is.
Going back to infinite dimensions, once my microphone gets the sample $\{x(iT/N) \, \mid \, 1 \leq i \leq N\}$, what is a good way of playing back the audio? We could try something like \begin{equation*} \hat{x}(t) = \sum_{i =0}^{N-1} a_{i} x(iT/N) f_{i}(t) \end{equation*}
for some fixed linearly independent functions $f_{i}$ and coefficients $a_{i}$. How should I choose the coefficients $a_{i}$? The functions $\{f_{0},\dots,f_{N-1}\}$ generate a finite dimensional subspace of my "signal space" (whatever that may be, but say it's a Hilbert space). One way to choose the $a_{i}$ is to try minimizing the "energy" (distance in the Hilbert space): choose $(a_{0},\dots,a_{N_1})$ so as to minimize \begin{equation*} \|\hat{x} - x\|_{\mathcal{H}}^{2}. \end{equation*} One of the first things we learn about Hilbert spaces is how to find this $\hat{x}$. (Of course, this is only one way to approximate a function, but this seems to be one of the classical approaches in signal processing.)