I am trying to determine whether $C[a,b]$, the set of all continuous functions defined on the interval $[a,b]$ with the inner product $\langle x(t),y(t)\rangle=\int_a^bx(t)y(t)dt$ is separable or not.
I have tried to find a numerable set that is dense, but I did not succeed. My idea was to find, for any continuous function, a sequence of functions contained in such set that converges to it. However I am struggling with the fact that the norm is the one that comes from the inner product.
How could I approach the problem?
Thanks in advance.
By Weierstrass approximation theorem, every continuous function can be approximated by a polynomial in $\sup$ norm. Then it is not hard to see that the polynomial can be approximated by a polynomial with rational coefficients in such a norm. All these polynomials form a countable set $\mathcal{D}$
Now pick a polynomial $p\in\mathcal{D}$ such that $\|f-p\|_{\infty}<\epsilon$, then $\|f-p\|^{2}=\int_{a}^{b}|f-p|^{2}dx\leq(b-a)\|f-p\|_{\infty}^{2}<(b-a)\epsilon^{2}$.