How Can Different Definitions of Separating Families of Seminorms Be Reconciled?

[julian]

In the context of families of seminorms I've come across these two definitions;

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

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

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 \{ p_I \} is separating if for each pair of points x \not= y we find p_I such that p_I (x) \not= p_I (y)

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

 

[rasmhop]

(iii) is not the same kind of separation as that in (i) and (ii). To see the difference just take the real line and let the family of seminorms be the set consisting of just the ordinary absolute value norm. Then clearly (i) holds because |x|=0 implies x=0, but (iii) does not hold because 1\not= -1, but |1|=|-1|.

In fact if you take as a family of seminorms just a single seminorm p, then (i) or (ii) holds precisely if p is a norm, but (iii) only holds when the underlying vector space consists of just 0 (otherwise pick non-zero x and we get x\not= -x, but p(x)=p(-x)).

Separation is a nice intuitive geometric concept so the term has been overloaded to mean different things in different contexts (just as other terms like regular, normal, space, linear, etc. mean many things).

Your condition (iii) is only usually used when we need our family of functions to tell the difference between any pair of elements, but in a vector space this is not necessary because if our family tells us that there is some kind of separation between 0 and x-y, then there must also be separation between x and y by linearity, but as the seminorms themselves are not linear it is way too strong to expect to say that elements are the same if they have the same norm. You just want to say that elements are separated if their difference has a different seminorm than 0.

In particular you may think about the real line (or plane, or \mathbb{R}^n) where we cannot tell the difference between 1 and -1 with our norm, but the way to think about it is that we want to translate and instead tell the difference between 0 and 2.

You may think of condition (i) and (ii) as being generalizations of the axiom of a norm that |x|=0 implies x=0.

In slightly more technical terms we sometimes call a topological space separated if it is Hausdorff. A locally convex topological vector space is given precisely by a vector space with a family of seminorms which induces the topological structure. The resulting topological vector space is Hausdorff if and only if it is induced by a separating family of seminorms.