Let $H$ be a Hilbert Space and let $T \in BL(H, K)$ be surjective then prove that the adjoint of $T$ is bounded below.
Since $T$ is surjective, so for any $y \in K, \exists$ $x \in H$ such that $Tx = y$ $\implies ||y||=||Tx|| \leq ||T||.||x|| \implies ||y|| \leq c||x||$, for some $c>0$.
We also define a unique $T^*: K \to H$ such that $\langle Tx, y\rangle = xT^*y$, as the adjoint of $T$.
It would be very helpful if I could get an insight of how to proceed from here. Thanks.