Hi,

Let $\displaystyle E$ be a Hilbert space and let $\displaystyle A: E \to F$ be a linear, bijective and compact operator. Prove that $\displaystyle B: F \to E, B = A^{-1}$ is not continuous.

Hint: $\displaystyle (e_n)_n$ is an orthonormal and total sequence in $\displaystyle E$ which convergens weak to $\displaystyle 0$ then $\displaystyle Ae_n \to 0$ as $\displaystyle A$ is compact.

I don't know how I can use the hint?

Anyone?

Thanks in advance.