= Solution
As explained by <admissible polynomials for a singular inverse-linear weight>, there is a domain issue: the constant <polynomial> does not belong to the weighted space, because $\int_0^1x^{-1}dx=\infty$. Hence the full space $\Pi_{n-1}$ is not a subspace of $V$, and the preceding inner-product projection theorem cannot be applied to it as stated.
Nevertheless a meaningful finite-error <best approximation> does exist after restricting to admissible <polynomials>. The target $x^n$ lies in $V$ for every $n\ge1$, since its squared integrand near zero is $x^{2n-1}$. A <polynomial> $p\in\Pi_{n-1}$ has finite error exactly when $p(0)=0$: otherwise the residual tends to a nonzero constant and its weighted squared integral diverges; if $p(0)=0$, it has at least a simple zero and is <integrable> in the required norm. Thus the admissible approximation space is
$$
\boxed{\Pi_{n-1}\cap V=\operatorname{span}\{x,x^2,\ldots,x^{n-1}\}.}
$$
For $n=1$ it is $\{0\}$, giving <best approximation> zero. For $n\ge2$ it is a finite-dimensional inner-product subspace with positive definite <Gram matrix>, so it has a unique <orthogonal projection> of $x^n$. Therefore \b[the full <polynomial>-space formulation is not defined inside $V$, but the finite-error interpretation has a unique <best approximation>]. Declaring that no <best approximation> can exist under any interpretation would miss this admissible subspace.
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