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Suppose that $a_m$, $m \in \mathbb{N}$, is a sequence of bounded linear operators on a Hilbert space converging strongly to an bounded linear operator $a$. If U is a finite-dimensional subspace of H, is it true that the orthogonal projections $P_{a_m(U)} : H \rightarrow a_m(U)$ converge strongly to the orthogonal projection $P_{a(U)}$?

My impression is that this is true. However, I'm unable to prove it. Any help would be appreciated.

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