Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose that we have a Hausdorff locally convex space with its topology $\tau$ and let $P(X)$ be a separating family of $\tau$-continuous semi-norms so that $\tau$ is generated by $P(X)$. How do we prove the following:

If $p_1,\cdots,p_n\in P(X)$, then we can find $p\in P(X)$ and a constant $k\ge 1$ such that for each $i\in \{1,\cdots,n\}$ and each $x\in X$, we have $p_i(x)\le k\cdot p(x)$.

share|improve this question
1  
Are there other assumptions about $P(X)$? for example, if $X$ is the space of continuous functions on $[0,1]$ and $\rho_x(f):=|f(x)|$, it gives a separating family of semi-norms. –  Davide Giraudo Nov 9 '12 at 17:14

1 Answer 1

A semi-normed space doesn't need to have the mentioned property. For example, take $C[0,1]$, the space of continuous functions with real values on the unit interval, and the semi-norms given by $\rho_x(f):=|f(x)|$. If $x_j,1\leqslant j\leqslant n$ are different elements of $[0,1]$ and $t\in [0,1]\setminus \{x_1,\dots,x_n\}$, we can find a continuous function which is $1$ at $x_1$ and $0$ at $t$.

However, if we assume the set of semi-norms filtrating, that is, for each finite collection of semi-norms $\{p_{i_1},\dots,p_{i_n}\}$, we can find an element $p_{i_{n+1}}$ such that for each $x\in X$, $\max_{1\leqslant j\leqslant n}p_{i_j}(x)\leq p_{i_{n+1}}(x)$.

For a vector space with semi-norms $\{p_i,i\in I\}$, we can add new semi-norms by the following way: for $F\subset I$ finite, define $$\rho_F(x):=\max_{i\in F}p_i(x).$$ This gives a family of semi-norms which gives the same topology as $\{p_i,i\in I\}$.

share|improve this answer
    
Thx. So my question would then be obvious if I assumed that the family $P(X)$ is filtrating. Am I right? –  juniven Nov 10 '12 at 7:30
    
Anyway, our $X$ here refers to any locally convex space whose topology is determined by a separating family $P(X)$ of continuous seminorms on $X$. I guess, with this definition of $PX)$, it follows that P(X) is always filtrating. Am I right? –  juniven Nov 10 '12 at 7:41
    
Yes, when the family is filtrating, there is no problem. What do you mean by continuous in the second comment? –  Davide Giraudo Nov 10 '12 at 10:54
    
I mean that every semi-norm on $X$ is continuous with respect to the topology in the locally convex space $X$. –  juniven Nov 10 '12 at 12:48
    
So, going back to the family $P(X)$, does it follow that it is filtrating? –  juniven Nov 10 '12 at 14:36

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.