Let $X$ be a normed space, $X^*$ its dual space and $\left\|{T}\right\|=\sup\{|T(x)|:\left\|x\right\|=1\}$, the usual norm in $X^*$. Let $(T_n)\subseteq X^*$.
It is true that if $T_n\to 0$ in $X^*$ then $T_n$ converges pointwise to zero, because for every $x\in X$ we have $|T_n(x)|\le \left\|{T_n}\right\|\left\|{x}\right\|$.
The converse is not true. However, in all counterexamples I've seen we have $\dim X=\infty$. Could it be true if $\dim X<\infty$? If not, could anyone help me to find a counterexample even in this case?
Thank you.
When $X $ is infinite-dimensional, the topology of pointwise convergence in $X^*$ can never agree with the topology of norm-convergence. Because the former is weak$^*$-convergence. And in the weak$^*$ topology, the unit ball is compact. In the norm topology, though, the unit ball is compact precisely when $X $ is finite-dimensional.