Suppose we are given a Hilbert space A, an infinite dimensional closed subspace B and an inner product $<\cdot,\cdot>$. A sequence of inner products $<\cdot,\cdot>_i$ converge to $<\cdot,\cdot>$ in the sense that $C^{-1}||v||\le||v||_i\le C ||v||,\forall i,v$ and for any $v,w\in A$, $<v,w>_i\rightarrow <v,w>$ as $i\rightarrow \infty$. Can we prove that for any $v\in A$, the orthogonal projection to B with respect to $<\cdot,\cdot>_i$, $P_i(v)$ converges to $P(v)$ (defined similarly)? If not, what additional conditions are required?
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(My current progress is shown here.)
If for any $v\in A$,
$$\sup_{|w|=1}|\langle v,w\rangle _i-\langle v,w\rangle |\rightarrow 0,$$
then $P_i(v)$ converges to $P(v)$ weakly.
If
$$\sup_{|v|=|w|=1}|\langle v,w\rangle _i-\langle v,w\rangle |\rightarrow 0,$$
then $||P_i-P||\rightarrow 0$.
These are routine arguments. I am not sure if there're stronger results.
