I have some trouble with proper understanding of $H_0^1$ space. I confess that I am now beginning to study functional analysis and maybe my question may seem rather trivial. However I would like to know if $$H_0^1(\Omega)$$ where $\Omega$ is an open bounded subset of $\mathbb R^N$, is a finite dimensional Hilbert space.
It is not finite dimensional.
One way of seeing this is if you consider the simple case $\Omega = ~ ]0,1[ \subset \Bbb R$. Then $H^1_0(\Omega)$ is the set of square integrable functions, whose derivative is also square integrable, and that the trace map on the boundary is $0$.
A subset of this is the set of continuous functions on $]0,1[$ such that vanish at $0$ and $1$. This set is infinite dimensional: think Fourier transform.
In fact, $H^1_0(\Omega)$ is a separable Hilbert space. This means that it admits a countable Hilbert basis.