Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is **not** orthogonal. Define $V_n = \text{span}(w_1, ..., w_n)$.

Is it possible to find an operator $Q_n:V \to V_n$ satisfying

- $(Q_n v-v, v_n)_H = 0 $ for all $v \in V$ and $v_n \in V_n$
- $\lVert Q_n v \rVert_V \leq C\lVert v \rVert_V$ where $C$ is independent of $n$

?

This kind of stuff comes up in Galerkin approximations but usually the basis used is orthogonal wrt. $H$ and orthonormal wrt. $V$. In my situation, I cannot use such a nice o.g/o.n basis because it is preferable to use a different special basis, and so the task is much harder. (When the basis is orthogonal in $V$ and o.n in $H$, the standard proof of this uses compactness of $V \subset H$ and Hilbert-Schmidt theory to get the result 2. after defining $Q_n$ as satisfying 1.)

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