Finite dimensional approximations of operators on Hilbert spaces Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$.  Let $A:V \to V$ be an operator.  Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n \to V$ be the injection so that $i_n^\dagger$ is the orthogonal projection.  Finally, define $A_n = i_n^\dagger \circ A \circ i_n : V_n \to V_n$.
1)  Are necessary and sufficient conditions known for the spectrum of $A_n$ to converge to the spectrum of $A$?
2)  Same question, but for the eigen-spaces?
(p.s.  I am an engineer with a fair knowledge of differential geometry.  I apologize if this question is trivial.  Functional analysis is a weakness for me.)
 A: $A$ is in the norm closure of finite dimensional operators iff $A$ is a compact operator.
Then the spectrum and the eigenspaces of $A_n$ converge to that of $A$.
A: This is a very natural question, of course! In 1900, this would have been a hot topic!
But, as we find, and as @PeterMichor's answer indicates, the finite-dimensional "approximations" to a given operator do not approximate it in operator norm. I think it is reasonable to be surprised at this.
But the compelling significance of the question does not allow us to stop with a technical objection, in my sense of this. That is, we "should" ask what inferences can be made about the spectrum of perturbations of an operator. This turns out to be a highly non-trivial issue, but the answer is not "nothing".
In particular, for example, as in Kato's book on Perturbation Theory, compact perturbations are understandable.
In physics, "singular" perturbations of reasonable operators are understandable.
So, not knowing the larger context of the question, nevertheless one can reasonably say that while the spectrum of an operator is not usually related to that of its "finite-dimensional (bad-) approximations", useful things can be said.
Perhaps the questioner can clarify the context...
