The linear functional of a sequence is bounded, then the sequence is bounded.

Question

Let $V$ be a normed linear space, and $\{v_n\}_{n=1}^\infty$ be a sequence in $V$, $V'$ be the dual of $V$. If for any linear functional $l\in V'$, the sequence $\{l(v_n)\}$ is bounded. Show that the sequence $\{v_n\}$ is bounded in $V$.

Answer 1

Apply the principle of uniform boundedness (or Banach-Steinhaus) to the set of functionals $\{F_n : n\in\mathbb N\}$, where $F_n : V'\to\mathbb R$ is defined by $F_n(\ell) := \ell(v_n)$.

1. Show that each $F_n$ is bounded and that $\|F_n\|\le\|v_n\|$. Then use Hahn-Banach to show that actually $\|F_n\| = \|v_n\|$.
2. Use the principle of uniform boundedness to show that $\sup_n\|F_n\| < \infty$.