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

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

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