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Does there exist a metric on the set of all monotone functions defined on a given compact set in $\mathbb{R}^k$ and taking values in a given compact set in $\mathbb{R}^m$ such that convergence in this metric is equivalent to pointwise convergence?

Monotonicity in both domain and image is understood in the component-wise partial order sense.

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  • $\begingroup$ If $m>1$, please explain what you mean by "monotone functions". $\endgroup$ – uniquesolution Oct 17 '15 at 20:01
  • $\begingroup$ @uniquesolution, as indicated above, I mean the component-wise order. But let's for simplicity consider $k=1$ and $m=1$. $\endgroup$ – MerylStreep Oct 17 '15 at 20:02
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    $\begingroup$ In general, over large enough domain, no! It is locally convex though. That is already pretty nice feature as it comes very close to metric spaces already: The topology is induced by seminorms. But it may still miss topological properties as being first countable. $\endgroup$ – C-Star-W-Star Oct 17 '15 at 21:23
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    $\begingroup$ @MerylStreep: You might try to find criteria which guarantee that your topology is metrizable like: Urysohn's theorem, cardinality of seminorms, bounded neighborhoods. However, the first to pretty much will fail. For the last one consult Rudin, Functional Analysis. (There's a more readable version as PDF file on the internet from a different author though it is a plagiat!!) $\endgroup$ – C-Star-W-Star Oct 17 '15 at 21:32
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    $\begingroup$ @MerylStreep: Yep that's what I thought so: For locally convex spaces there's no big difference to metric spaces at least not for the matter of convenience. The topology is even defined w.r.t. seminorms: $\mu_x(f):=\|f\|_x:=|f(x)|$ ('mu': Minkowski gauge!) $\endgroup$ – C-Star-W-Star Oct 17 '15 at 21:40

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