# Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same)

Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and continuously embedded in $H$. Let $\{b_j\}$ be a basis for $H$ and for $V$. Define $$P_n:H \to \text{span}(b_1,...,b_n)$$ by $$(P_n h-h,v_n)_H = 0\quad\text{for all v_n \in \text{span}(b_1,...,b_n)}$$ by truncation.

Is it true that $$P_n:V \to V$$ is a bounded operator where the constant that bounds it does not depend on $n$?

It is true as an operator from $H$ to $H$ (take $v_n = P_nh$ in the definition) but not sure for $V$.

• Why do you say that $P_n$ maps $V$ to $V$? If $b_1\notin V$, then $P_1$ takes values outside of $V$. – ˈjuː.zɚ79365 Jun 28 '13 at 14:24
• @ˈjuː.zɚ79365 sorry $b_i$ are in $V$ too. They're basis for both spaces. – michael_faber Jun 28 '13 at 18:57
• In what sense is $\{b_j\}$ a basis? It can't be orthonormal in both inner products. – Nate Eldredge Oct 11 '13 at 16:02