 Solution

The Riesz representation theorem says that for every bounded linear functional $f$ on a Hilbert space $H$, there is a unique $z\in H$ such thatand $\Vert f\Vert =\Vert z\Vert$.

For $T\in L(H)$ and fixed $y\in H$, the mapis a bounded linear functional, withRiesz therefore gives a unique vector, denoted $T^{\ast }y$, such thatfor every $x$. Uniqueness in the representation theorem shows that $y\mapsto T^{\ast }y$ is linear. Moreover,so the adjoint operator belongs to $L(H)$ and $\Vert T^{\ast }\Vert \le \Vert T\Vert$.

If $(e_j)$ is an orthonormal basis and $A=(a_{ij})$ is the matrix of $T$, then the matrix of $T^{\ast }$ isits entries are $\overline{a_{ji}}$.