= Solution
A polynomial $p_n\in\Pi_n$ is a best <polynomial least-squares approximation> when it minimizes the weighted error <norm> over all polynomials of degree at most $n$:
$$
\boxed{\|f-p_n\|\leq\|f-q\|\quad\text{for every }q\in\Pi_n.}
$$
Equivalently it minimizes $\int_0^1x^{-1/2}(f(x)-q(x))^2\,dx$. The weight is integrable, so the expression is finite for continuous functions; it defines a genuine <norm> because a continuous function with zero weighted squared integral must be identically zero.
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