I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without mention of convergence or function spaces), the Laplace and Fourier transforms, and the sampling theorem.
I still don't truly understand the sifting property, or why sampling is multiplication with a shifted delta instead of a shifted 1 (which would preserve the value of the signal at that point right?). I'm still not sure how to determine when a signal's Fourier series will converge, when you are allowed to take a Fourier or Laplace transform, or whether the derivations of the transform properties we learned are mathematically justified and rigorous.
This led me to self study books like Tolstov's Fourier Series and Richards's Theory of Distributions, which I could not follow due to my non-rigorous math background. The Osgood Stanford lectures were quite helpful, but I still feel quite sad that I do not have a deeper understanding of my math tools.
According to my school's EE graduate course catalog, there are a couple of classes covering Hilbert, Banach, and L^p spaces, and linear functionals. Is it possible to cover these topics without a formal mathematical background? Should I begin self learning real and complex analysis to prepare for these topics?
Forgive me for this lengthy question. Thanks!