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Recall that the open mapping theorem is true for Banach spaces. Furthermore, it is also true for Frechet spaces, which are complete metric spaces(with extra properties). It makes me wonder whether we can say that open mapping theorem applies to all complete metrizable topological vector spaces, since the key ingredient Baire category theorem certainly applies to all complete metric spaces and the norm in the proof of Banach spaces can be adapted to metric hopefully.

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  • $\begingroup$ The Banach space theorem applies to linear maps... $\endgroup$ – David C. Ullrich Nov 26 '17 at 21:23
  • $\begingroup$ I am sorry, I should certainly say a complete metrizable topological vector space. $\endgroup$ – Keith Nov 26 '17 at 21:24
  • $\begingroup$ Any completely metrizable locally convex topological vector space is Frechet. $\endgroup$ – Justthisguy Nov 27 '17 at 3:43
  • $\begingroup$ That is true, but what about complete metrizable topological vector space? It is not necessarily locally convex. $\endgroup$ – Keith Nov 27 '17 at 3:53

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