Let $X,Y$ be normed spaces and $T_n:X\to Y$ a sequence of continuous linear functions. It is easy to see that
$\left\|{T_n}\right\|_{\infty}\to 0$ implies $\left\|{T_n}\right\|_*\to 0$
because if $\sup\{\left\|{T_n(x)}\right\|:x\in X\}\to 0$ then of course $\sup\{\left\|{T_n(x)}\right\|:\left\|{x}\right\|=1\}\to 0$.
I'm trying to find an example to show $\left\|{T_n}\right\|_*\to 0$ does not imply $\left\|{T_n}\right\|_{\infty}\to 0$. I know $X$ must be infinite dimensional but I can't find it yet.
Any hints?
Thank you.