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)