## eigenvalues of a compact positive definite operator!!!

Let A be a compact positive definite operator on Hilbert space H.
Let ψ1,...ψn be an orthonormal set in H.
How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where
λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order.
Can someone give me a hint???
 Mentor Both the left and right expression look like tr(A).
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Try induction. Do you know that $$\lambda_1=sup\{~\vert~x\in H,~\|x\|=1\}$$ ?? If you know this, then the case n=1 should be easy. Can you find an argument to deal with the other cases?