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Let $X$ be a normed space. Suppose $E$ is a subset of $ X^*$ (The space of continuous linear functionals). For every $\phi\in E$, define seminorm $p_\phi: X\to [0,\infty)$ such that $p_\infty (x)= |\phi(x)|$. If $\tau$ is the topology generated by these seminorms and $(X,\tau)$ is locally convex space then $(X,\tau)^* = span(E)$.

I do not have any idea about it. Please help me. Thank so much.

niki
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  • One idea: by the definition of $\tau$, all the elements of $E$ are continuous, so all the linear combinations of elements of $E$ are continuous. – Martín-Blas Pérez Pinilla Nov 12 '14 at 18:48
  • @Martín-BlasPérezPinilla : I think We should show that every $\phi\in (X,\tau)^*$ is $|.|-$continuous and conversely. From your idea, I think you claim every element of $E$ is $\tau -$ continuous. – niki Nov 12 '14 at 18:59
  • In this case there is no norm. And $(X,\tau)^*$ is the dual with respect to $\tau$. – Martín-Blas Pérez Pinilla Nov 12 '14 at 19:03
  • @Martín-BlasPérezPinilla : I edited it. – niki Nov 12 '14 at 19:04
  • Since by assumption, $E$ is a subset of the space of norm-continuous linear functionals, it is clear that every $\lambda \in \operatorname{span} E$ is norm-continuous. By definition of $\tau$, it is elementary that every $\lambda\in \operatorname{span} E$ is $\tau$-continuous. The not-quite-trivial part, and the one you need to spend a little work proving is that every $\tau$-continuous linear functional on $X$ belongs to $\operatorname{span} E$. – Daniel Fischer Nov 12 '14 at 19:10
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    This could be helpful: http://math.stackexchange.com/questions/709781/intersection-of-kernels-implies-linear-dependence-of-functionals – PhoemueX Nov 12 '14 at 19:28
  • @PhoemueX : Thanks for your helpful comment. Now I can prove it. – niki Nov 12 '14 at 19:33

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