I'm having trouble with the definition of (pre)compact sequences in the space $H^{-1}(\Omega)$ understood as the dual space of the Sobolev space $W_0^{1,2}(\Omega)$, and $\Omega$ a domain in $\mathbb{R}^n$.
Say we have a sequence $\{ f_n \}$ in $H^{-1}(\Omega)$, so continuous linear functions $f_n : W_0^{1,2}(\Omega) \to \mathbb{R}$, I have two questions.
- I've found in Evans PDE the definition of a precompact sequence as
A sequence is precompact if it has a convergent subsequence.
Just like when we are talking about precompact and compact sets is there a notion for compact sequences? If so, which is the difference? I've found the term in some papers/texts (about compensated compactness) but I don't know if they are talking about the same thing as Evans.
- Say we have a condition like this
$$ \left| f_n(\phi) \right| \leq C_n \| \phi \|_{W_0^{1,2}}, $$
for every $\phi \in W_0^{1,2}(\Omega)$ where $C_n \to 0$ and doesn't depend on $\phi$, does this imply that the sequence is (pre)compact? If so, why?
Edit: What if we just had $\left| f_n(\phi) \right| \to 0$ instead?