So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this.
- I know 'big ell 2' and 'little el 2' are vector spaces. However I need a little bit more understanding on what these are. The lecture defined them ast $l^2 := e^n(m) = \begin{cases} 0 & \mbox{if } m \neq n \\ 1 & \mbox{if } m = 1 \end{cases} $ and $ L^2 := V=\{ f_x : [-\pi, \pi] \rightarrow \mathbb{C} | \frac{1}{2\pi}\int_{-\pi}^\pi {|f_x|^2 dx} < \infty \}$. Can you explain more about these vector spaces, why they are important, a wiki page, and basis why/how the basis are orthonormal and infinite dimensional.
- Since they(what is "they") are orthonormal $f_x$ from $L^2$ wraps "something" around the unit circle.$f_x $ is defined as $f_n(x) = e^{inx}$ and for every x, $|e^{inx}| = 1$
- Enough about them, lecture notes says Fourier series connects $L^2([-\pi,\pi])$ to $l^2(\mathbb{Z})$ using orthognal projection. I am very confused about this.
The reason I am asking here is that I can't find a good resource (pdf, wiki) that conects linear algebra to fourier series. Theres plenty of information on either subject but not for both.