Consider the Sobolev space $W^{1,2}([0,1])$.
If I am not mistaken this space admits an orthonormal basis:
$1, 2\pi n\sqrt{2} \cos(2\pi n x), 2\pi n\sqrt{2} \sin(2\pi n x), \quad n= 1,2,...$
Now I impose two constrains, namely $u(0)=0$ and $u(1)=0$, $\forall u \in W^{1,2}([0,1])$.
Can you provide an orthonormal basis for the new constrained space?
In general, what can be said about this new space?
Thanks a lot