There is some sequence $x_n$ in $C[a,b]$ such that $\|x_n\| =1$ but $x_n(t)$ tends to $0$ for all t $\in [a,b] $.

Question

Let $\| \|$ be a norm on $C[a,b]$. Then there is some sequence $x_n$ in $C[a,b]$ such that $\|x_n\| =1$ but $x_n(t)$ tends to $0$ for all t $\in [a,b] $.

Answer 1

Example $$ x_n(t)=\left\{\begin{array}{lll} 0 & \text{if} & t\in \big[0,\frac{1}{n}\big], \\ nt-1 & \text{if} & t\in \big[\frac{1}{n},\frac{2}{n}\big], \\ 3-nt & \text{if} & t\in \big[\frac{2}{n},\frac{3}{n}\big], \\ 0 & \text{if} & t\in \big[\frac{3}{n},1\big]. \end{array} \right. $$ Then $x_n$ continuous, $0\le x_n(t)\le 1$ and $x_n\big(\frac{2}{n}\big)=1$.