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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 $V$.

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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 $\sup_n\|F_n\| < \infty$.
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  • $\begingroup$ Is it possible that you can give me more hints? I'm still confused. $\endgroup$ Oct 17 '17 at 4:14
  • $\begingroup$ Don't you know this principle? $\endgroup$
    – amsmath
    Oct 17 '17 at 4:15
  • $\begingroup$ Thank you for your hints. I know the theorem but I was not sure how to use it. Thank you! $\endgroup$ Oct 17 '17 at 4:22

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