# Show {$x^k$}$_{k\geq 0}$ is complete in $L^2[a,b]$

I'm thinking of using the Weierstrass Approximation Theorem where the span of monomials is dense in the continuous functions.

• That works fine. – Lord Shark the Unknown Feb 27 at 5:21
• I'm not too sure however to move past that step. Is the continuous function dense in integrable functions? – HCS Feb 27 at 5:35
• polynomials are $L^\infty$-dense in $C[a,b]$ and so $L^2$-dense in $C[a,b]$. $C[a,b]$ is $L^2$-dense in $L^2[a,b]$. – Lord Shark the Unknown Feb 27 at 6:06