Density on a specific functional space.

Question

I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let $$ \mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{R}^n)\times\mathbb{R}^m $$ and consider the following subset of $\mathcal{X}$: $$ \mathcal{D}\colon=\left\{(\epsilon, \varphi)\in\mathcal{X}\colon \epsilon(0)=C\varphi+G\epsilon(1)\right\} $$ where $C$ and $G$ are given matrices with suitable dimensions. I'd like to prove whether $\mathcal{D}$ is dense in $$ \mathcal{L}^2(0,1;\mathbb{R}^n)\times\mathbb{R}^{m} $$ endowed with the following "standard" inner product:

$$ \langle(f_1,f_2),(g_1, g_2)\rangle=\int_{0}^1f_1(x)g_1(x)dx+f_2^\top g_2 $$ Thanks!

Answer 1

I think $\mathcal D$ is dense: Let $f\in L^2([0,1],\mathbb{R}^n)$, $\varphi\in\mathbb{R}^m$ and $\delta>0$. Take $g\in C^1([0,1],\mathbb{R}^n)$ such that $\|f-g\|_{L^2}<\delta$. Let $\psi=C\varphi+Gg(1)$. Now modify $g$ in a small neighbourhood of $0$ to get some $h\in C^1([0,1],\mathbb{R}^n)$ such that $\|g-h\|_{L^2}<\delta$ and $h(0)=\psi$. Then $(h,\varphi)\in \mathcal D$ and $\|(h,\varphi)- (f,\varphi)\|< 2\delta$.