= Solution
Use the convention that the <inner product> is linear in its first argument, as required by the displayed representation of a <linear functional>. Choose an <orthonormal basis> $e_1,\ldots,e_n$, for instance by <Gram-Schmidt orthogonalization>. Define
$$
v=\sum_{j=1}^n\overline{f(e_j)}e_j.
$$
For $u=\sum_ju_je_j$, conjugate linearity in the second argument gives
$$
\langle u,v\rangle=\sum_ju_jf(e_j)=f(u).
$$
Thus
$$
\boxed{f(u)=\langle u,v\rangle\quad\text{for every }u\in V.}
$$
The representation is unique as well: if two vectors represent the same functional, their difference is orthogonal to every vector and therefore to itself, so it is zero.
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