Finding the conjugate transpose of a finite dimensional matrix in $M_{n \times n} (\mathbb{C})$ is easy enough, but what about the infinite dimensional case with a linear operator $T$ on some infinite dimensional vector space?
My linear algebra textbook states that the conjugate transpose of $T^{*}$ of $T$ is the transformation satisfying $\langle T(x) , y \rangle = \langle x , T^{*}(y) \rangle$ for all $x,y \in V$.
I can't seem to make sense of how I would use this in practice however. Any insight would be greatly appreciated.
