If you are interested in developing some intuition on Hilbert spaces, I recommend learning some Fourier analysis. Thinking about how vibrating strings conceptu

If you are interested in developing some intuition on Hilbert spaces, I recommend learning some Fourier analysis. Thinking about how vibrating strings conceptually have infinitely-many modes, but the higher frequency modes matter less and less, gets across a lot of the intuition. The Fourier integral and the spectrum of light is also intuitive.