Show that the projection from $C[0,1]$ onto $P_n[0,1]$ under $L^q$ norm is nonlinear when $q>1$ and $q\neq 2$.

Question

Let $C[0,1]$ denote the space of real valued continuous functions defined on $[0,1]$. Let $P_n[0,1]$ the space of real polynomials with degree not greater than $n$ defined on $[0,1]$. Let $q>1$ denote a finite real number. For any $f\in C[0,1]$, one can show by compactness argument and uniform convexity of $L^q$ norm that there exists an unique $P(f)\in P_n[0,1]$ such that
\begin{equation}
||f-P(f)||_{L^q(0,1)}=\inf\limits_{p\in P_n[0,1]}||f-p||_{L^q(0,1)}.
\end{equation}
In this fashion, we define a projection operator: $f\in C[0,1]\mapsto P(f)\in P_n[0,1]$. How to show that $P(\cdot)$ is nonlinear when $q\neq 2$? Could anyone help me show this? I really don't know how to start it.

Answer 1

I am just giving a (possibly a trivial) perception here. Let $q=m$ an even positive integer. \begin{equation} \|f-\sum_{i=0}^{n} a_i x^i\|_m^m = \int_{0}^{1} (f-\sum_{i=0}^{n} a_i x^i)^m dx\\ = \sum_{i_1,...,i_{n},i_{m+1}} \frac{m!}{i_1!...i_{n+1}!} (-1)^{i_1+...+i_m} a_1^{i_1}...a_n^{i_n} \int_{0}^{1} f^{i_{n+1}}x^{\sum_{k=0}^n ki_k} dx \end{equation}

for $m=2$, if we differentiate the above equation w.r.t $a_i$ and set to $0$, then its a system of linear equations in $f,a_1,...,a_n$ and for $m >2$, if we differentiate above and set to $0$, its a set of non-linear equations in $f,a_1,...,a_n$.