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.)



Finite dimensional approximations of operators on Hilbert spaces
$A$ 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 $A$.


