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In Reed & Simon's text on functional analysis they give the following proposition:

Every locally convex space has a directed family of seminorms equivalent to the family defining the space

with the following proof:

If $\{\rho_\alpha\}_{\alpha \in A}$ defines the space, let $B$ be the set of finite subsets of $A$. If $F \in B$, let $d_F = \sum_{\alpha \in F} \rho_\alpha$. Then $\{d_F\}_{F \in B}$ is directed and equivalent to the initial set.

I see why $\{d_F\}$ is directed, since for $F, G \in B$, $$d_F(x) + d_G(x) \leq Cd_{F\cup G}(x)$$ for some constant $C$. This follows since seminorms are nonnegative and $F \cup G$ is finite. However, I could not figure out why the topology induced by this family of seminorms is equivalent to the initial set.

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All of the original seminorms are in the directed family, so the topology induced by the directed family contains the original topology.

Also, each seminorm in the directed family is continuous with respect to the original topology, so we didn't manage to produce any new open sets.

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    $\begingroup$ Ah I see, because the only "new" seminorms are finite sums of seminorms that were in the original set, these "new" seminorms are also continuous in the original topology. Is that right? $\endgroup$
    – CBBAM
    May 29 at 2:24
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    $\begingroup$ Right! (character limit) $\endgroup$
    – diracdeltafunk
    May 29 at 2:24
  • $\begingroup$ Got it, thank you for your help! $\endgroup$
    – CBBAM
    May 29 at 2:25

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