Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?

  • $\begingroup$ These "nice properties" are shared with normed linear spaces. Other "nice properties" are too strong to provide wide applications, e.g. local compactness implies finite dimensionality. So in a sense there is nothing more than these nice properties, but perhaps that should be enough? $\endgroup$ – hardmath Aug 28 '14 at 10:56
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    $\begingroup$ Well, one of the nice properties is that a version of the Hahn-Banach theorem still holds, which is of course one of the most important tools in functional analysis. $\endgroup$ – Vincent Boelens Aug 28 '14 at 13:14
  • $\begingroup$ Also, for some applications the norm topology is simply too restrictive, so one should consider weaker topologies. The ones that work best are often locally convex, in part because of the Hahn-Banach theorem. $\endgroup$ – Vincent Boelens Aug 28 '14 at 13:17
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    $\begingroup$ What you describe is not motivation. It doesn't make anyone care about such spaces. The motivation is from examples of important spaces in analysis that are not Banach spaces, such as the Schwartz space. $\endgroup$ – KCd Aug 31 '14 at 14:04
  • $\begingroup$ @KCd What are other spaces that motivate this definition? $\endgroup$ – user109871 Oct 5 '18 at 20:00

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