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If $X$ is a normed space than it is well known that its norm topology and its weak topology coincide if and only if $X$ is finite-dimensional.

Now I asked myself the same question about general locally convex spaces and found the following answer at Math.SE: https://math.stackexchange.com/a/138906/7110

But the problem is that though I now have an equivalent characterization of when the weak and initial topology coincide, I still don't have any non-trivial (i.e., not a normed space) example of such a space (there seems to be no example given in the paper that proves the equivalent characterization).

(So I want to rule out that the paper is talking about something vacuous, i.e., so that we don't have an equivalent characterization of something that does not exist [in the non-normed case].)

So what is a concrete example of a locally convex space (which is not a normed space) such that its weak and initial topology coincide?

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  • $\begingroup$ I also asked the opposite question to this one here: mathoverflow.net/q/156540/13356 $\endgroup$
    – AlexE
    Commented Feb 3, 2014 at 9:17
  • $\begingroup$ The simplest such example is the space often denoted by $\phi$---consisting of those sequences of reals for which only finitely many terms fail to vanish. More generally, any space which is the locally convex inductive limit of its finite dimensional subspaces, in particular a direct sum of finite dimensional spces $\endgroup$
    – alpha
    Commented Feb 3, 2014 at 15:46
  • $\begingroup$ @alpha: Could you please elaborate a bit more on your comment and / or give a proof of the statement? If you write it as an answer, I will gladly accept it. $\endgroup$
    – AlexE
    Commented Feb 4, 2014 at 12:33
  • $\begingroup$ I gave an answer here: mathoverflow.net/questions/156540/… $\endgroup$
    – Sergei Akbarov
    Commented May 26, 2014 at 14:24
  • $\begingroup$ @AlexE The term "initial topology of $X$" is a bit ambiguous to me: will you clarify it? thank you! $\endgroup$
    – Pietro Majer
    Commented Jul 9, 2018 at 17:42

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