I try to understand the Fourier transform using linear algebra, but something goes wrong
Below is my attempt:
Fourier transform is a process try to orthogonal project the input signal $x(t)$ to infinite many basis (like $e^{jwt}$), but what is the inner product defined here? the coefficient for each basis can be proven to be $\langle x(t),e^{jwt}\rangle$ divided by $\langle e^{jwt},e^{jwt}\rangle$ , where $\langle f(t),g(t)\rangle$ denoted as inner product of $f(t)$ and $g(t)$