We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$. The infimum of $L$ on $A_N$ is denoted $a_N = \inf\{L(P) \mid P \in A_N\}$. (a) Show that the infimum of $L$ on $A_{N}$ is attained. In what follows, we denote by $B_{N}$ the set of $P \in A_{N}$ such that $L(P) = a_{N}$. (b) Show that $B_{N}$ is a convex compact subset. (c) Verify that $B_{N}$ contains an even polynomial.
We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$. The infimum of $L$ on $A_N$ is denoted $a_N = \inf\{L(P) \mid P \in A_N\}$.
(a) Show that the infimum of $L$ on $A_{N}$ is attained.
In what follows, we denote by $B_{N}$ the set of $P \in A_{N}$ such that $L(P) = a_{N}$.
(b) Show that $B_{N}$ is a convex compact subset.
(c) Verify that $B_{N}$ contains an even polynomial.