= Solution
Define the <finite-rank operator>
$$
\boxed{P_nx=\sum_{i=1}^n x_i^*(x)x_i.}
$$
It is linear and bounded, with $\|P_n\|\leq\sum_{i=1}^n\|x_i^*\|\|x_i\|$. Biorthogonality gives exactly the two required values on $x_j$. It also gives
$$
P_n^2x=\sum_{i=1}^n\sum_{j=1}^n x_j^*(x)x_i^*(x_j)x_i=P_nx.
$$
Thus \b[$P_n$ is a bounded projection] onto $\operatorname{span}\{x_1,\ldots,x_n\}$, with kernel $\bigcap_{i=1}^n\ker x_i^*$. Its action on the dense span determines it uniquely among <bounded linear operators>. These are the partial-sum projections; no uniform bound in $n$ has yet been assumed.
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