Let $$C_c^\infty([a,b],\mathbb{C}):=\{f\in C^\infty([a,b],\mathbb{C}): f^{(k)}(a)=f^{(k)}(b)=0, k=0,1,...\}$$ Then $C_c^\infty([a,b],\mathbb{C})$ is dense in the Hilbert Space. Can anyone give me some tips in order to prove this theorem? How can one approach this problem?
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I assume your Hilbert space to be $L^2[a.b]$, then here is a proof. It works even for $L^p(\mathbb R^n)$. – Martin Argerami May 08 '20 at 02:29
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@MartinArgerami my functions are defined in the Complex plane, isn't it a problem? Or one can argument via the isomorphism of \C to \R^2? – riemannfanboy May 08 '20 at 14:54
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A function $[a,b]\to\mathbb R^2$ is harder to deal with than a function $[a,b]\to\mathbb C$. In the proof I mentioned, every single step works with a complex-valued function; no changes needed. – Martin Argerami May 08 '20 at 15:11