= Solution
Assume first that $\|P_n\|\leq M$ for all $n$. Given $x\in X$ and $\epsilon>0$, choose a finite linear combination $y$ of the $x_i$ with $\|x-y\|<\epsilon/(M+1)$. For all sufficiently large $n$, $P_ny=y$. Therefore
$$
\|P_nx-x\|\leq\|P_n(x-y)\|+\|y-x\|\leq(M+1)\|x-y\|<\epsilon.
$$
Thus $P_nx\to x$, and the formula in part (iii) yields the norm-convergent expansion
$$
\boxed{x=\sum_{i=1}^\infty x_i^*(x)x_i.}
$$
This proves the stated existence of coefficients.
Conversely, suppose every $x$ has some norm-convergent expansion $x=\sum_i\alpha_ix_i$. Applying the continuous functional $x_j^*$ to its partial sums and passing to the limit gives $\alpha_j=x_j^*(x)$. Hence those partial sums are precisely $P_nx$, so $P_nx\to x$ for every $x$. In particular $\sup_n\|P_nx\|<\infty$ for each fixed $x$. The <Uniform boundedness principle>, using completeness of $X$, now gives
$$
\boxed{\sup_n\|P_n\|<\infty.}
$$
This proves the reverse implication. The coefficients are automatically unique, so the equivalence is the <uniform partial-sum criterion for a Schauder basis> for the given minimal total <sequence>. Its projection bound is the <basis constant>.
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