If the set is defined as
$$\{a_0+a_1x+a_2x^2+…+a_nx^n, \text{where }n\ge 0 \text{ and } a_0+a_1+…+a_n=0\},$$
is the set dense in $C[0,1]$ and $C[-1,0]$?
For the first question, I'm thinking using Weierstrass theorem that there exists a function $f$ belongs to $C[0,1]$ s.t $p_n$ converges uniformly to $f$ in $[0,1]$. And $p_n(1)$ converges uniformly to $f(1) = 0$ and the set belongs to the new set $\{f|f(1)=0\}$ which is not $C[0,1]$. So it's not dense in $C[0,1]$. Is the idea right?
For the second question, I'm thinking proving the set is dense in $C[-1,1]$ and therefore dense in $C[-1,0]$. But I'm wondering is it possible for a set dense in $C[-1,1]$ but not in $C[0,1]$.
Best Answer
Your conclusion that this set of polynomials is not dense in $[0,1]$ is correct, but the argument is not quite right. Instead, you should choose a particular function $f\in C[0,1]$ with $f(1)\neq 0$ such that no sequence of admissible polynomials converges to $f$. Try, for instance, $f(x)=1$. The argument would hinge on the fact that no sequence of polynomials can converge pointwise to $1$ when $x=1$.
For the second question, your argument won't work. The set of polynomials is not dense in $C[-1,1]$ by the same argument you just gave. A set that is dense in $C[-1,1]$ will always have a dense set of restrictions to $C[0,1]$.