In the context of families of seminorms I've come across these two definitions;(adsbygoogle = window.adsbygoogle || []).push({});

i) a family of seminorms [itex]\{ p_I \}[/itex] is separating if [itex]p_I = 0[/itex] for all [itex]I[/itex] implies [itex]x=0[/itex].

ii) for a family of seminorms, when for every [itex]x \in X / \{ 0 \}[/itex] there is a seminorm [itex]p_I[/itex] such that [itex]p_I (x) > 0[/itex].

It is easy to show these imply each other. I have now come across another definition for a family of functions to be separating (not necessarily seminorms):

iii) a family of functions [itex]\{ p_I \}[/itex] is separating if for each pair of points [itex]x \not= y[/itex] we find [itex]p_I[/itex] such that [itex]p_I (x) \not= p_I (y)[/itex]

It is easy to show iii) implies ii) if you assume the functions of condition iii) satisfy [itex]p_I (0) = 0[/itex]. My question is how condition iii) could be implied by either of the others.

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# Functions that separate points

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