Let $V$ be subspace of $\ell^2$ which contains all 1 summable sequences. For each natural number $n$, define $T_n: V \to \mathbb R$ by $T_n(x)=\sum_{i=1}^n x_i$. Then $T_n$ is not uniformly bounded on unit ball $\|x\|_2\leq1$.
My intuition says it has something to do with closed and bounded in infinite dimensional banach space need not be compact. But I don't know how to get a firm answer. Could you please tell me the reason? Thank you very much for your time.
Basically you need to find $x$ such that $\lVert x \rVert_2 \leq 1$ but $\lvert T_n(x) \rvert \to \infty$. Put explicitly you require $x$ to satisfy $$ \sum_{i=0}^{\infty} x_i^2 \leq 1 \quad \text{and} \quad \sum_{i=1}^{\infty} x_i = \infty$$ Can you think of such $x$?