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The question is:

Let $X$ be a normed linear space and let $B$ be a dense subset of $X^*$(the conjugate space of $X$). If a sequence $\{x_n\}$ in $X$ is bounded, and if $\lim_n x^*(x_n)$ exists for each $x^*\in B$, then prove that $\lim_n x^*(x_n)$ exists for each $x^*\in X^*$.

Could anyone tell me the way to solve this problem? Thanks!

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Hint: it suffices to show that $x^*(x_n)$ is Cauchy. –  user27126 Feb 26 '13 at 18:22
$|x^*(x_n)-x^*(x_m)|\le |x^*(x_n)-x^*_k(x_n)|+|x^*_k(x_n)-x^*_k(x_m)|+|x^*_k(x_m)-x^*(x_m)|$? –  user60610 Feb 26 '13 at 18:33
Basically yes. But then you need to specify how you choose your $k$, and how large $n,m$ have to be. –  user27126 Feb 26 '13 at 21:33

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