This is exercise 12.16.15 from Dieudonné's treatise on analysis volume 2
It attempts to find necessary and sufficient conditions on a sequence of control points in I = [0,1] for the Lagrange interpolation polynomials to converge uniformly, for each continuous function $f : I\to R$.
exercise 12.16.15 https://i.stack.imgur.com/tRzXR.png
My ideas ... $P_n$ is linear and $\|P_n(f)\| \le (\sum_{i=0}^{n} \|q_{in}\| )\|f\| $ so is continuous.
$(\Rightarrow)$ if $\forall f \in \mathscr{C}(I)$ we have $P_n(f) \rightarrow f$ in $\mathscr{C}(I)$ , then for each $f \in \mathscr{C}(I)$ , $\sup_{n} \|P_n(f)\| < \infty$ . By 12.16.4 below $(P_n)$ is therefore equicontinuous , so by 12.15.7.1 below $\|P_n\|$ is bounded .
$(\Leftarrow)$ I can't show ... $\|P_n\|$ bounded allows us to conclude by 12.15.7.1 below that $(P_n)$ is an equicontinuous sequence in $\mathscr{L}( \mathscr{C}(I) , \mathscr{C}(I) )$ . It seems we're supposed to infer somehow for a given $f \in \mathscr{C}(I)$ that the sequence $(P_n(f))$ of maps in $\mathscr{C}(I)$ is equicontinuous. If we could infer that, and if we make additional assumptions on the sequence of control points $(a_{ij})_{i \le j}$ , for example that $\{ a_{ij} : j \in N , i \le j \}$ is dense and $j_1 \le j_2$ implies $\{ a_{ij_1} : i \le j_1 \} \subset \{ a_{ij_2} : i \le j_2 \}$ , then we can conclude $(P_n(f))$ converges simply on a dense subset and so by 7.5.5 below converges simply in $I$, therefore by 7.5.6 below uniformly, which would complete the proof.
But I see no way to infer that for a given $f \in \mathscr{C}(I)$ that the sequence $(P_n(f))$ of maps in $\mathscr{C}(I)$ is equicontinuous.
theorems 12.16.4 and 5 theorem 12.15.7.1 theorem 7.5.5 theorem 7.5.6
all in this image ... https://i.stack.imgur.com/EzBgx.png
