= 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 that
$$
f(x)=\langle x,z\rangle\qquad(x\in H),
$$
and $\lVert f\rVert=\lVert z\rVert$.
For $T\in L(H)$ and fixed $y\in H$, the map
$$
f_y(x)=\langle Tx,y\rangle
$$
is a bounded linear functional, with
$$
|f_y(x)|\leq\lVert T\rVert\lVert x\rVert\lVert y\rVert.
$$
Riesz therefore gives a unique vector, denoted $T^*y$, such that
$$
\langle Tx,y\rangle=\langle x,T^*y\rangle
$$
for every $x$. Uniqueness in the representation theorem shows that $y\mapsto T^*y$ is linear. Moreover,
$$
\lVert T^*y\rVert=\lVert f_y\rVert
\leq\lVert T\rVert\lVert y\rVert,
$$
so the <adjoint operator> belongs to $L(H)$ and $\lVert T^*\rVert\leq\lVert T\rVert$.
If $(e_j)$ is an orthonormal basis and $A=(a_{ij})$ is the matrix of $T$, then the matrix of $T^*$ is
$$
A^*=\overline A^{,T};
$$
its entries are $\overline{a_{ji}}$.
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