sincerely, I'm stuck. Then, I have two questions:
if we take $V=\{v\in H^1(0,1) ; v(0)=0\}$ and $Q=\{ w_1,w_2\}$ is a lineary independent set where $w_1 = \frac{*}{\Vert *\Vert_{V\cap H^2(0,1)}}$ and $w_2 = \frac{**}{\Vert **\Vert_{H^1(0,1)}}$
My first question is:
Can we constract a basis in $V\cap H^2(0,1)$ and $H^1(0,1)$ represented by $\{w_1,w_2,....,w_n\}$ linearly independent subset of $Q$ (using the Gram Schmidt orthonormalization process)?
My second question is:
Let us suppose $u_0 \in H^1(0,1)$ . are we can find for all $\epsilon >0$, $u \in V\cap H^2(0,1)$ such that
$\Vert u-u_0 \Vert_{H^1(0,1)} < \epsilon $
$H^2(0,1)={u\in H^1(0,1) / \frac{\partial^2 u}{\partial^2 x}\in L^2(0,1)}$
– Richard Feb 03 '16 at 11:56