Is the space of tempered distributions 1st countable ?

In summary, the answer to the question is that the space of tempered distributions with the weak-* topology is not first countable.
  • #1
dextercioby
Science Advisor
Homework Helper
Insights Author
13,366
3,505
Hi everyone,

the question is simple: is [itex] \mathcal{S}'\left(\mathbb{R}^3\right) [/itex] a first countable topological space ?

I have no idea, honestly. (The question has occurred to me from a statement of Rafael de la Madrid in his PhD thesis when discussing the general rigged Hilbert space formalism. He says that even though the space of wavefunctions is assumed 1st countable, his antidual generally isn't. So I took the simplest case of a rigged Hilbert space: [itex] \mathcal{S}\left(\mathbb{R}^3\right)\subset \mathcal{L}^2 \left(\mathbb{R}^3\right)\subset \mathcal{S}^{\times}\left(\mathbb{R}^3\right) [/itex])
 
Physics news on Phys.org
  • #2
dextercioby said:
the question is simple: is [itex] \mathcal{S}'\left(\mathbb{R}^3\right) [/itex] a first countable topological space ?

Assuming the weak*-topology my recollection is that this space is not first countable.
 
  • #3
The only handy reference I have on weak*-topology is this: http://en.wikipedia.org/wiki/Weak-star_operator_topology and has to do with operator spaces and particularly the space of the trace-class operators... How's that related to the (anti)dual of the Schwartz space ?
 
  • #4
Last edited:
  • #5
Alright, thank you, got that. So can you show it's not first countable?
 
  • #6
If memory serves the argument is essentially that the space of tempered distributions with the weak-* topology is not metrisable (since for topological vector spaces the two are equivalent). I forget how exactly this is shown, but if you perform an internet search I am sure this argument will turn up somewhere.
 
  • Like
Likes 1 person
  • #7
It doesn't show as the proof, only as the result, thing which is quite frustrating.

I found an answer on the competitor's website, for those of you interested in the same question.
Thanks jgens for the interest shown.
 
Last edited:

FAQ: Is the space of tempered distributions 1st countable ?

What is the definition of a 1st countable space?

A 1st countable space is a topological space in which for every point, there exists a countable neighborhood base. This means that there is a countable collection of open sets that contain the point and every open set containing the point contains one of the sets in the collection.

How does the concept of 1st countability relate to tempered distributions?

Tempered distributions are a class of functions that are used to generalize the concept of a function in the context of Fourier analysis. The space of tempered distributions, denoted by S', is a topological vector space. The question of whether this space is 1st countable is important in understanding the properties and behavior of tempered distributions.

Why is it important to know if the space of tempered distributions is 1st countable?

The concept of 1st countability is important in topology as it helps to classify and understand different types of topological spaces. Knowing whether the space of tempered distributions is 1st countable can provide insight into the convergence of sequences of tempered distributions and the properties of their limits.

Is there a simple way to determine if the space of tempered distributions is 1st countable?

No, there is not a simple way to determine if the space of tempered distributions is 1st countable. It requires a deep understanding of topology and functional analysis to make this determination. However, there are some general theorems and techniques that can be applied to certain cases to determine 1st countability.

Are there any applications of knowing whether the space of tempered distributions is 1st countable?

Yes, there are various applications of understanding the 1st countability of the space of tempered distributions. For example, it can help in the development of numerical methods for solving partial differential equations, which often involve the use of tempered distributions. It can also aid in the study of generalized functions and their properties.

Similar threads

Replies
16
Views
2K
Replies
11
Views
2K
Replies
3
Views
2K
Replies
2
Views
4K
Replies
2
Views
4K
Replies
2
Views
4K
Replies
10
Views
3K
Back
Top