Let $H$ be a Hilbert space. Given any family $\{g_j,h_j\}^{n}_{j=1} \subset H$, and for any $f \in H$.
Let $A_nf=\sum_{j=1}^{n}\langle f,g_j \rangle h_j$
If $X$ is a closed subspace in $H$, then there exists an orthogonal projection $P$ such that $rangeP=X$, $kerP=X^\bot$.
Can we conclude $A_n$ is bounded and an orthogonal projection by the fact that $\{\sum_{j=1}^{n}\langle f,g_j \rangle h_j:\forall f\in H\}$ is a closed subspace in $H$?
Any help would be appreciated.
Best Answer
Being an orthogonal projection is much more than just having a closed range. For starters, if $P$ is an orthogonal projection, then $P^2=P$. But the given operators $A_n$ in general don't satisfy this property (unless we know something special about the elements of the family $\{g_j,h_j\}^{n}_{j=1}$). Or another consequence of being an orthogonal projection is that $\|P\|=1$, which again isn't true here in general (unless … see above).