Let $w_i$ be a basis for a Banach space $V$. We have $V \subset H$ a continuous and dense embedding into a Hilbert space $H$.
Define $y_m = \sum_{i=1}^m a_{im}w_i$. We have that $y_m \to y$ in a $H$ as $m \to \infty$.
Why is it true that $|y_m|_H \leq C|y|_H$?
I don't know why it holds. I found this is Lions "Optimal Control of Systems" beginning at page 100. Also Wloka claims it in page 403.