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 $L(P) = \int_{-1}^{1} P(x)\,dx$. (a) Verify that $A_{N}$ is a convex subset of $\mathbb{R}_{N}[X]$. (b) Show that the expression $$\|P\|_{1} = \int_{-1}^{1} |P(x)|\,dx$$ defines a norm on $\mathbb{R}_{N}[X]$. (c) Show that $A_{N}$ is closed in the normed vector space $\left(\mathbb{R}_{N}[X], \|\cdot\|_{1}\right)$.
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 $L(P) = \int_{-1}^{1} P(x)\,dx$.
(a) Verify that $A_{N}$ is a convex subset of $\mathbb{R}_{N}[X]$.
(b) Show that the expression
$$\|P\|_{1} = \int_{-1}^{1} |P(x)|\,dx$$
defines a norm on $\mathbb{R}_{N}[X]$.
(c) Show that $A_{N}$ is closed in the normed vector space $\left(\mathbb{R}_{N}[X], \|\cdot\|_{1}\right)$.