# Definition talk:Topological Vector Space

The topology on $V$ is not inherited from $K$ - this only really makes sense in finite dimensions ie if $V = K^n$. When we say $\cdot : K \times V \to V$ should be continuous, we mean as a map from $\struct {K \times V, \tau_K \times \tau_V}$ (where $\tau_K \times \tau_V$ is the product topology of $\tau_K$ and $\tau_V$) to $\struct {V, \tau_V}$. So $\tau_V$ is dependent on $\tau_K$ somehow. I went through the verbosity on the other talk page. Caliburn (talk) 13:44, 18 February 2023 (UTC)

- This complexity is currently swept under the rug, making it difficult to understand what is going on. Perhaps the fault is with me not having studied the right books. --prime mover (talk) 14:43, 18 February 2023 (UTC)

## Hausdorff requirement

I've got myself in a bit of a twist regarding the Hausdorff requirement. One nice way you can prove that a vector space with a topology is a TVS is to prove vector addition and scalar multiplication are continuous, and then that the singleton $\set { {\mathbf 0}_X}$ (or any singleton) is closed. However this is an awkward equivalence to put up, because to prove this criterion we have to deploy results about Hausdorff topological vector spaces that still hold when the Hausdorff property is relaxed. I feel the easiest way to do this is to have Definition:Non-Hausdorff Topological Vector Space, (we want to read "non-Hausdorff" as "not necessarily Hausdorff" rather than "not Hausdorff") cite this in articles as (not necessarily Hausdorff) topological vector space, change over pages (that I will need) that only need the continuity of operations like Translation of Open Set in Topological Vector Space is Open, and then prove the criterion. Thoughts? Caliburn (talk) 09:49, 16 July 2023 (UTC)

- An alternative would be moving to "Hausdorff Topological Vector Space". Don't
*really*fancy doing this but it is an option. Caliburn (talk) 09:57, 16 July 2023 (UTC)- This latter is the most immediate way I would do it. Big job, I'll set up a rename template and refactor advanced template. --prime mover (talk) 10:52, 16 July 2023 (UTC)
- Would it be alright if I added a template to each page stating that each TVS page is under review? There are probably nearly or over 100 TVS results (of which I would guess about 10 use the Hausdorff property) that need to be reviewed, and I prefer this to keeping a personal list since we can have a disclaimer that pages are under review. Caliburn (talk) 08:25, 17 July 2023 (UTC)

- This latter is the most immediate way I would do it. Big job, I'll set up a rename template and refactor advanced template. --prime mover (talk) 10:52, 16 July 2023 (UTC)

- That sounds like a plan. There may be a template already built which may be appropriate -- if not, feel free to craft one. --prime mover (talk) 20:25, 17 July 2023 (UTC)