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Let X be a Banach space. By $C([0,T];X)$ I denote all continuous functions on the compact interval $[0,T]$ of values in $X$. How to prove that $C([0,T];X)$ dense in $L^2([0,T];X)$ ? Can I use the same argument as in Understanding denseness of $C^\infty$ in $L^p$ space.?

zorro47
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  • @Vessemir...Do you use theLebesgue measure to this question? Because i can give you a proof. – Marios Gretsas Aug 08 '17 at 18:15
  • @yanko this is only "true" (in the sense of a vector of polynomials) for finite dimensional $X$. Still it might be extended on separable spaces by some fancy construction, but it does not cover full generality. [original comment removed] – Max Aug 08 '17 at 18:20
  • @Max, this result still holds for infinite dimensional Banach spaces $X$. The integration can all be understood using the Bochner integral, and this is how I interpret the $L^2([0,T];X)$ space in my answer below. – Yousuf Soliman Aug 08 '17 at 18:33
  • @yousufsoliman my comment refers to another comment that has been deleted, someone claimed it would be obvious by polynomial density which is clearly not true. – Max Aug 08 '17 at 18:47
  • @Max, ahh my bad. I missed that! – Yousuf Soliman Aug 08 '17 at 18:47

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First show that the Bochner integrable simple functions are dense in $L^p([0,T];X)$ - you can do this easily using the Lebesgue dominated convergence theorem for the Bochner integral. Now you just need to show that $\mathscr{C}([0,T];X)$ is dense in the space of Bochner integrable simple functions: let $I\subseteq[0,T]$ be a measurable set and let $\{\varphi_n\}_{n=1}^{\infty}\subset\mathscr{C}([0,T];\mathbb{R})$ be a sequence of continuous (you can make these smooth if you want to) such that $\varphi_n\to\chi_E$ in $L^p([0,T];\mathbb{R})$. Now for any vector $x\in X$ we have $\varphi_n\cdot x\to \chi_E\cdot x$ as $n\to\infty$, and so by linearity the space of continuous functions is dense in the space of simple functions, and hence is dense in $L^p([0,T];X)$.

Yousuf Soliman
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  • Of course you need some regularity on the measure which you are performing the integration with respect to if you want to find the sequence ${\varphi_n}$ above, i.e. this is all possible if the measure on $[0,T]$ is a Radon measure - in particular this holds for the Lebesgue measure. – Yousuf Soliman Aug 08 '17 at 18:31