Will the following sets be convex in the real space $C[a, b]$:
a) polynomials of degree $n$;
b) polynomials of degree not higher than $n$.
A subset $M$ of $\mathbb{R}^n$ is called convex if it contains, along with any pair of its points $(x,y)$ also the entire segment $[x,y]$:
$$x,y\in M,0\leq\lambda\leq 1\implies \lambda x + (1-\lambda)y\in M.$$
A polynomial of degree $n$ in a single variable $x$ can be written in general form
$$a_nx^n+a_{n–1}x^{n–1}+\dots+a_2x^2+a_1x+a_0=\sum a_i x^i$$
To be honest, I don’t even know where to start. Please give me a hint. And what is the difference between a) and b)? Thank you.
a) NO. For example take $x^n$ and $-x^n$ then $0=\dfrac{1}{2}(x^n-x^n)$ is not the polynomial of degree $n$.
b) Yes. Notice that $deg(f+g)\leq max\{deg(f),deg(g)\}$.