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.

The definitions I am using are as follows:

A vector space $V$ equipped with a family $P$ of semi-norms such that $\cap_{p\in P}\{x\in V : p(x) = 0\} = \{0\}$ is called a locally convex vector space.

I am concerned with the topology given by the collections of subsets $U$ of $V$ such that for all $x\in U$ there exists $n\geq 1$ and $p_{1}, ... , p_{n}$ in $P$ and $\epsilon_{1}, ..., \epsilon_{n} > 0$ such that

$x\in \{y\in V : p_{i}(x - y) < \epsilon_{i}$ for $i = 1,... n\}\subset U$.

Showing this topology is closed under finite intersection is very straight-forward. But arbitrary unions is eluding me because if I take a family of such sets, say $\{U_{\alpha}\}$, then I don't get a finite subset of seminorms and epsilons to use to show that the resulting union is still open.

How can I finish this?

share|improve this question
2  
But... You're describing a base for the topology given by the sets of the form $U_{p_1,\ldots,p_n} (x) = \{y \in V : p_i (x-y) \lt \varepsilon_i \text{ for } i = 1,\ldots,n\}$! A set is open if and only if it is a union of such sets. –  t.b. Jan 12 '12 at 5:11
1  
Yes I am describing the topology in terms of the basis. I tried to verify the properties of a basis but that turned into an absolute mess (in particular with the 2nd basis axiom) so I went back to following the way my professor did it in the notes, which was just show that the resulting open sets U form a topology. However he left the union axiom as an exercise. –  Kyle Schlitt Jan 12 '12 at 5:24

1 Answer 1

up vote 2 down vote accepted

Each $x$ in the union must be in one of the particular sets $U$ appearing in the union, and you get your $n$, $p_i$s and $\varepsilon_i$s from this $U$.

share|improve this answer
2  
Thank you. This is definitely a facepalmer. Next time I am going to have to try doing some jumping-jacks to clear my head before I waste 20 minutes typing up a question to which the answer is this trivial. I was thinking about the problem totally backwards. –  Kyle Schlitt Jan 12 '12 at 5:26

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.