The weighted orthogonal-polynomial Galerkin projection is well-defined

Each numerator is finite by [F3], and each denominator $h_j$ is finite and strictly positive by [F1]–[F2]. Thus every coefficient in the displayed finite sum is a defined real number.

The integrability hypotheses and [F1] make every term in the finite sum integrable, and [F4] distributes its integral. Orthogonality of the $p_j$ then gives, for each $i\le N$, $\int (u-{\mathrm{\Pi }}_{N}u)p_{i}d\mu =\int u{p}_{i}d\mu -{\sum }_{j=0}^N\frac{\int u{p}_{j}d\mu }{h_j}\int p_j{p}_{i}d\mu =0,$ because all terms with $j\mathrm{\ne }i$ vanish and the $j=i$ term equals $\int u{p}_{i}d\mu$. Thus ${\mathrm{\Pi }}_{N}u\in V_N(\mu )$ has the required residual orthogonality.

If $v\in V_N(\mu )$ has the same residual property, then $w=v-{\mathrm{\Pi }}_{N}u$ lies in $V_N(\mu )$ and is orthogonal to every basis vector, hence to itself. Positive definiteness in [F2] gives $\int w^{2}d\mu =0$, so $w=0$ in $L^2(\mu )$ and $v={\mathrm{\Pi }}_{N}u$. This proves uniqueness.