Polynomials of best approximation

Question

The question is about approximating the continuous function in an interval $[a, b]$. If we consider the linear space of all such functions endowed with the norm

$$||f|| = \max_{x \in [a, b]}|f(x)|$$

then the best error approximating a function $f$ using polynomials $p \in \pi_n$ (with $\pi_n$ is denoted the set of all polynomials of degree $\leq n$) is

$$E_n(f) = \inf_{p\in \pi_n} ||f-p||$$

I need to show that $E_n(f + g) \leq E_n(f) + E_n(g)$. I don't know how to approach the problem. Any hints?

Answer 1

Let $$E_n(f)=a,\qquad E_n(g)=b$$ There exist two polynomials $p_1$ and $p_2$ such that $$\|f-p_1\|<a+\epsilon,\qquad \|g-p_2\|<b+\epsilon$$ so $$E_n(f+g)\le\|f+g-p_1-p_2\|\le a+b+2\epsilon$$ for all epsilon, so $$E_n(f+g)\le a+b=E_n(f)+E_n(g)$$

Answer 2

Hint: if $p$ is a good approximation to $f$ and $q$ is a good approximation to $g$, try approximating $f+g$ by $p + q$.