Ufyukyu

1 $begingroup$ Not for homework, I am trying to self study functional analysis and encountered the following problem. We let $C[0,1/2]$ the continous functions defined on that subset of the real line. We look at a subspace of $C[0,1/2]$ , consisting of all polynomials on $[0,1/2]$ and call it $W$ . Given $delta>0$ , set. $g(x)=deltasum_{n=1}^{infty}frac{1}{n+1}x^n$ Where $xin[0,1/2]$ . First, we are tasked with showing that $g$ is in the open ball $B(0,delta)$ . I suppose this is done by showing that the norm of $g$ must be less than delta (we are using the supremum norm inherited from $C[0,1/2]$ ). But from that result, we are tasked with using it to conclude that $W$ cannot be an open subset of $C[0,1/2]$ . So I must show that for some $win W$ , there is no $epsilon$ , such that an open ball $B(v,e