Definition talk:Weak Closure
The notation is not the clearest.
When we are setting up definitions like this, I believe it is important and instructive to specify everything in as much detail as possible. At the moment it is confusing.
We have that $X$ is a topological vector space, and then the $\struct {X, w}$ is a topological vector space with a topology $w$ imposed.
But $X$ itself is properly defined as $\struct {V, \tau}$ where $V$ is a vector space and $\tau$ is a topology.
And then we have $F$ described as being a topological field.
What exactly is the underlying set and what exactly is the underlying vector space and what exactly is the relation between the topology on $X$ and the topology $w$?
I would sort of expect to see:
- $F$ would be specified as $\struct {F, \tau}$ where $F$ is tacitly understood as being shorthand for the field $\struct {F, +, \times}$, where $F$ is the underlying set of the whole thing.
- $V$ is a vector space over the field $F$, such that $X = \tuple {V, \tau}$ is the resulting topological vector space over the underlying vector space $V$
and then
- $\tuple {X, w}$ is the topological vector space over the underlying vector space $V$ and so $X = \struct {V, w}$
or:
- $\tuple {\tuple {V, \tau}, w}$
which would need further explaining.
There was a lot of discussion way back concerning whether it was always necessary to say something like:
- Let $T = \struct {S, \tau}$ be a topological space
rather than just taking the topology for granted:
- Let $T$ be a topological space
but then we've lost track of what the underlying set is.
I still suggest that it is nearly always better to use the full definition whenever the relationship between these objects is subject to misunderstanding. --prime mover (talk) 11:51, 18 February 2023 (UTC)
- It is true that a topological vector space $\struct {X, \tau}$ is also a topological vector space with its weak topology - though this needs a proof (you need to show that vector addition and scalar multiplication are still continuous operations in the weak topology) and I haven't put it up yet. It's fine just talking about $\struct {X, w}$ as a topological space for now, I haven't defined "weakly x" for any property x that requires the space to be a topological vector space.
- In as much verbosity as is reasonable, you would have a topological field $\struct {K, \tau_K}$, with the operations of $K$ understood as $+_K$ and $\circ_K$, (which $\tau_K$ makes continuous) then you would consider a vector space $\struct {X, +_X, \circ_X}_K$. Then we give $X$ a Hausdorff topology $\tau_X$, depending on $\tau_K$, such that the maps $+_X : \struct {X, \tau_X} \times \struct {X, \tau_X} \to \struct {X, \tau_X}$ and $\circ_X : \struct {K, \tau_K} \times \struct {X, \tau_X} \to \struct {X, \tau_X}$ are continuous. (where here products of topological spaces are understood with the product topology) Now consider the set of linear functionals $f : \struct {X, \tau_X} \to \struct {K, \tau_K}$ that are continuous, which is defined to be the "topological dual" written $\struct {X, \tau_X}^\ast$ - though we might want to emphasise the role of the topology on $K$ too. We then define the "weak topology" $w$ as the initial topology on $X$ with respect to the family of linear functionals $\struct {X, \tau_X}^\ast$ - the least topology making these functionals continuous. This depends on everything we've written so far: the operations on $X$, $K$ and the topologies $\tau_X$, $\tau_K$.
- One slight inconsistency that I might as well note is that topological groups/rings/fields are not required to have Hausdorff topologies, but topological vector spaces are. This then requires a bit of awkward wording, and means some theorems aren't in their absolute fullest generality. For example we have to say that if $K$ is a Hausdorff topological field, then $K^n$ is a topological vector space with the product topology given by $\tau_K$. If the topology on a topological vector space is not required to be Hausdorff, it is true that if $K$ is a topological field then $K^n$ is a topological vector space. Maybe we could define a "non-Hausdorff Topological Vector Space" for this, or remove the Hausdorffness requirement. Caliburn (talk) 13:39, 18 February 2023 (UTC)
- That is exactly the sort of verbosity I would expect.
- It might be a useful step to find a source work and see what that has to say. None of the definitions in this area are sourced, which is something that needs to be addressed. --prime mover (talk) 14:41, 18 February 2023 (UTC)
- The problem is that as we get further away from "standard undergraduate content", (this is stuff found in second or third courses in functional analysis) the more terse textbooks tend to be. It seems unlikely to me that a text would spell out individually what "weakly closed", "weakly compact" etc. means rather than just saying "defined in the obvious way", though I will have a look. Caliburn (talk) 12:07, 19 February 2023 (UTC)
- Gotcha. Maybe we need to exploit the resources of StackExchange or whatever else it out there. --prime mover (talk) 17:33, 19 February 2023 (UTC)