Weak convergence implies norm inequality.

Question

When I was reading Mathematical Methods in Quantum Mechanics With Applications to Schrodinger Operators by Gerald Teschl. Link here: http://www.ams.org/bookstore-getitem?item=gsm-157.

I found in page 56 that if $\psi_n \rightharpoonup \psi$ on a Hilbert space $H$ then $$\lim\inf \langle\psi,\psi_n\rangle \leq \|\psi\|\lim\inf \|\psi_n\|$$ And I don't know why. It looks like Fatou's theorem, but not exactly. What am I missing here?

Answer 1

This is just Cauchy-Schwarz. For any $n$ $$|\langle\psi,\psi_n\rangle|\leq\|\psi\|\|\psi_n\|.$$ Taking the $\liminf$ of both sides gives the result.