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This question comes from Linear and Nonlinear Functional Analysis with Applications (Philippe G. Ciarlet), Chapter 4, Problem 4.3-4.

4.3-4 Let $\mathcal{P}_n[0,1]=\left\{\left.p\right|_{[0,1]} ; p \in \mathcal{P}_n\right\}$, where $\mathcal{P}_n$ denotes the space of all polynomials $p: \mathbb{R} \rightarrow \mathbb{R}$ of degree $\leq n$, and let a number $q>1$ be given.
(1) Show that, given any function $f \in \mathcal{C}[0,1]$, there exists a unique polynomial $P f \in \mathcal{P}_n[0,1]$ such that $$ \|f-P f\|_{L^q(0,1)}=\inf _{p \in \mathcal{P}_n(0,1]}\|f-p\|_{L^q(0,1)} . $$ (2) Show that the mapping $P: \mathcal{C}[0,1] \rightarrow \mathcal{P}_n[0,1]$ defined in this fashion is linear if and only if $q=2$ (the proof of the "if" part is similar to that of Theorem 4.3-1(e)).

I have proved (1), my question is about how to prove (2) $"\Rightarrow"$ (only if) part. Any help is appreciated!

Hang
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    What specific ways have you tried for proving the said direction? Where're you stuck? – Ѕᴀᴀᴅ Dec 18 '22 at 02:14
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    Hint: the function $g(p):=(1/q)\Vert f - p \Vert^q$ is convex. Therefore, you can compute it's minimum by solving the system $dg(p,v) = 0$ for all $v$, where $dg(p,v)$ is the directional derivative of $g$ at $p$ in direction $v$. – Albe Dec 20 '22 at 19:10

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