Let $f \in \mathcal{C}([a,b])$ where $a$, $b$ are finite, and let $\{\phi_j\}$ be an increasing-in-degree orthogonal basis for the polynomials with respect to some weight function $w(x)\geq0$. The unique least-squares approximating polynomial exists and is defined by $$r_n^*(x) \equiv {\arg\min}_{p\in\mathcal{P}^n} \Vert f - p\Vert_{L_w^2}$$
It can be shown$^1$ that $$\lim_{n\rightarrow \infty} \Vert f - r_n^*\Vert_{L_w^2} = 0$$ but that this does not imply uniform convergence.
Could someone provide a counterexample that illustrates this property?
$^1$e.g., Thm. 4.7 An Introduction to Numerical Analysis, 2nd Ed.; Atkinson, 2004