In the following question we assume the eigenvalues are sorted in descending orders and the eigen-vectors and eigen-functions are sorted accordingly.
Given a symmetric $n\times n$ matrix $P$ and a perturbed one $P'=P+H$, how could we bound the distance of $||u_i-u_i'||$ using the spectral norm $||H||$ ? Wedin or Davis-Kahan theorem bounds the distance of eigen-space, my question is how to bound the $\ell_2$ distance of eigen-vectors?
Now if P and P' are self-adjoint operators in a Hilbert space, how could we bound $||u_i-u_i'||$ if we know the operator norm of $||H||$ ? $u_i$ and $u_i'$ are eigen-functions of the operators.