Is it true that the space $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? These are compactly supported functions that are $V$ valued, where $V$ is a Banach or Hilbert space.
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1$\begingroup$ Convolution with a smooth compactly supported bump function? $\endgroup$– André HenriquesCommented May 17, 2013 at 11:57
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3$\begingroup$ Isn't it superfluous to require compact support, given that the domain is a compact interval? $\endgroup$– MTSCommented May 17, 2013 at 13:16
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$\begingroup$ @MTS: $C_c^\infty(K) = \lbrace f\in C^\infty(\mathbb R): \mathrm{supp}(f) \subseteq K\rbrace$. $\endgroup$– Jochen WengenrothCommented May 17, 2013 at 13:47
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$\begingroup$ A related question: mathoverflow.net/questions/130276/mathcald0-tv-is-dense-in-w0-t $\endgroup$– András BátkaiCommented May 17, 2013 at 22:17
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1 Answer
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As Andre suggests: Convolution with smooth bump function with very small support will give you an approximation by a smooth function which however need not have support in $[0,T]$. However, you may first squeeze the support of the given function you want to approximate in order to make its support a compact subset of $(0,T)$. Then the support of the convolution stays in $[0,T]$ if the support of the bump function is small enough.
answered May 17, 2013 at 13:57