I am a beginner in the field of functional analysis and started with Conway's book. On the chapter of topological vector spaces, there is an exercise which I can't crack and would appreciate some help
- Let $X$ be a infinite dimensional vector space and let $\tau$ be the collection of all subsets $W$ of $X$ such that, if $x\in W$, then there is a convex balanced set $U$ with $x+U\subset W$ and $U\cap M$ open in $M$ for every finite dimensional linear manifold in $X$ (the topology in $M$ is the unique linear topology). a) $(X,\tau)$ is Locally convex space. b) a set $F$ is closed in $X$ if and only if $F\cap M$ is closed for every finite dimensional subspace. ...
The first part was ok, but the second part I am having trouble to prove the converse. So far, my conclusion is that b) is equivalent to $O$ is open iff $O\cap M$ is open for any finite dimensional subspace $M$. Which would imply that $\tau$ is the finest linear topology (since every subspace of a TVS is a TVS with induced topology and there is only one linear topology in finite dimensional spaces, right?!). If this makes sense, it means that the strongest linear topology is a locally convex topology, which makes me believe I am missing something here
I looked in some free online notes (http://www.math.uni-konstanz.de/~infusino/Lect12.pdf) and this topology is called the finest locally convex topology, and its characterization given in b) makes sense in countable dimensional TVS (also in the notes). On Conway's book however, no such countability is mentioned...
Anyways, I would appreciate some tips on how to solve this problem and further comments or references on the topic.
I believe you're right, that exercise in Conway is a mistake. You need countable dimension.
See Section II.6 in Schaefer's book on Topological Vector Spaces, also see Exercise 7 on page 69, which is similar to Conway's exercise but the countable dimension is required.