How to find the adjoint of a linear transformation between Hilbert spaces

Question

Let $T:H \to H$ be a bounded linear transformation between Hilbert spaces. How to construct its adjoint and how to prove that its adjoint is unique?

I know that $\langle Tx,y\rangle =\langle x,T^* y\rangle $ and I am wondering how to find the adjoint of a linear transformation in general.

Answer 1

for a fixed $y\in H$ the map : $$ \phi_y : H \to \mathbb{C} $$ defined by : $$ \phi_y(x)=\langle Tx, y\rangle $$ is a continuous linear functionals, in fact $$ |\phi_y(x)|=|\langle Tx, y\rangle |\leq \|Tx\|\|y\|\leq \|T\|\|y\|\|x\| $$ by Riesz representation theorem we can get a unique $Y\in H$ such that : $$ \phi_y(x)=\langle x, Y \rangle $$ the maps who match every $y$ to $Y$ is denoted by $T^*$, and verify : $$ \langle Tx, y\rangle =\langle x , T^* y\rangle $$