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Let $E$ be a locally convex Hausdorff topological vector space. Show that $E$ is isomorphic to a subspace of a product of normed spaces.

All I know is that, if $E$ is locally convex Hausdorff, then there is a family of separated semi-norms. The problem at this point is, I have no idea how to construct(if necessary) such a product of normed space and what this space looks like. Furthermore, I think maybe this has something to do with the family of separated semi-norms, but I don't know how to connect the separated pieces.

Any hints are welcomed.

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Furthermore, I think maybe this has something to do with the family of separated semi-norms

Very much so.

Let $E$ be a vector space (over $\mathbb{R}$ or $\mathbb{C}$ for simplicity), and $p \colon E \to [0,+\infty)$ a seminorm. Then

  • $\ker p := \{ x \in E : p(x) = 0\}$ is a linear subspace of $E$, and
  • $\hat{p} \colon E/\ker p \to [0,+\infty)$ given by $\hat{p}(x+\ker p) = p(x)$ is

    1. well-defined, and
    2. a norm on $E/\ker p$.

It should now not be hard to guess one appropriate product of normed spaces, and to verify all claims.

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  • $\begingroup$ Now I see, it becomes clear. Thank you for your help. $\endgroup$ – Frank Lu Feb 6 '15 at 20:59

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