Many times I've seen that Fourier series are justified because we are thinking that the set of all functions of the form $\sin(ax)$ and $\cos(ax)$ form a vector space. A function can therefore be expressed as some coefficient and my base vector.
But I just don't see why the set of all $\sin(ax)$ and $\cos(ax)$ form a vector space. For example, they don't satisfy the condition of closure under addition because when I add two of these I'll get a $\phi$ term and I won't have a function of the form $\sin(ax)$ or $\cos(ax)$.
Aside from Fourier series I had this problem where I had to prove that the space of all functions with form $\sin(ax)$ and $\cos(ax)$ where orthogonal on the interval $(-\pi,\pi)$. I don't think this is an actual vector space.
Thanks.