Dual Space Topology: A to B Inclusion Map

In summary, the conversation discusses a continuous inclusion map from A to B, where A and B are topological spaces. The conversation then moves on to the induced map between the topological dual spaces B^* and A^*, and asks if it is continuous and injective. The conversation looks at special cases and discusses how most notions tend to reverse under duality. The conversation then moves on to discussing the inclusion of dual spaces H^* in \Phi^* and why \Phi\subset H \implies H^*\subset\Phi^*. There is some confusion about the relation between the two spaces, but it is eventually clarified that the map from H^* to \Phi^* is injective and not surjective.
  • #1
kakarukeys
190
0
let
[tex]A \hookrightarrow B[/tex]
be a continuous inclusion map from A to B.
A, B are two topological spaces. [tex]A \subset B[/tex]

what can we say about the induced map between topological dual spaces
[tex]B^* \hookrightarrow A^* [/tex]?

is it continuous and injective?
 
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  • #2
Look at special cases. If we have an injective map

[tex]\mathbb{R}^2 \rightarrow \mathbb{R}^3[/tex]

then when we dualize, what sort of map do we get on the dual spaces:

[tex]\mathbb{R}^2^* \cong \mathbb{R}^2 \leftarrow \mathbb{R}^3 \cong \mathbb{R}^3^*[/tex]

?



Dualities tend to reverse most notions. "monic" and "epic" are dual notions, so when you dualize a monomorphism, it tends to become an epimorphism, and vice versa.


Of course, it's always a good idea to work out the details for yourself. :smile: It gives you good practice with the notions involved.
 
  • #3
thanks...
I know the answer now.
But it doesn't solve my problem of understanding certain steps in
http://en.wikipedia.org/wiki/Rigged_hilbert_space

"that is one for which the natural inclusion
[tex]\Phi\subset H[/tex]
is continuous. It is no loss to assume that [tex]\Phi[/tex] is dense in H for the Hilbert norm. We consider the inclusion of dual spaces [tex]H^*[/tex] in [tex]\Phi^*[/tex]."

Why [tex]\Phi\subset H \implies H^*\subset\Phi^*[/tex]?
Could you please take a look?
 
  • #4
I'm confused. You just said you understand why [itex]A \longmapsto B[/itex] implies [itex]B^* \longmapsto A^*[/itex]. So, I don't see why you don't get [itex]\Phi \longmapsto H[/itex] implies [itex]H^* \longmapsto \Phi^*[/itex].
 
  • #5
no... in my second post, I used set operation "[tex]\subset[/tex]"
I understand there is a map from [tex]H^*[/tex] to [/tex]]Phi^*[/tex] which is the induced map. The wiki articles says
[tex]\Phi\subset H \implies H^*\subset\Phi^*[/tex]

H* a subset of Phi* ? in what sense?
I mean the functions in H* and the functions in Phi* have different domains, how could the two be related?
 
  • #6
Set inclusion is a function. Duality reverses arrows, and hence inclusions. Also very few topoological spaces have any notion of duality.
 
  • #7
kakarukeys said:
H* a subset of Phi* ? in what sense?
I mean the functions in H* and the functions in Phi* have different domains, how could the two be related?
Ah, so that's what you're worried about.

Generally, injective maps are what matter, whether or not you have an actual subset is irrelevant. But inclusions are notationally convenient... so sometimes (for convenience) when we have an injective map, we identify the the objects of the domain with their images.

Basically, because the restriction map [itex]H^* \longmapsto \Phi^*[/itex] is so natural, there isn't really any benefit in making a distinction between an element of [itex]H^*[/itex] and its image.
 
  • #8
so is it right to say the wiki article is wrong?
the map from [tex]H^*[/tex] to [tex]\Phi^*[/tex] is onto but not injective.
So there is no inclusion and [tex]H^*[/tex] is not a subset of [tex]\Phi^*[/tex].
 
  • #9
Hrm. Now that I think more about it, I think the maps [itex]\Phi \rightarrow H[/itex] and [itex] H^* \rightarrow \Phi^*[/itex] are both monic and epic in the categorical sense. I know that epic doesn't always imply surjective... I don't know if monic implies injective in this category. :frown: (For a simpler example of what I mean, the ring homomorphism [itex]\mathbb{Z} \rightarrow \mathbb{Q}[/itex] is epic... but it's not surjective)

Oh well; if we want to prove injectivity, that just means we'll have to resort to a dirtier method. I think it's not too difficult to prove that the kernel of the map [itex] H^* \rightarrow \Phi^*[/itex] is zero.
 
  • #10
hi, thanks for your reply... I think I got the answer
the map [tex]H^* \longmapsto \Phi^*[/tex] is injective and not surjective.
it's (in general) not surjective because H is infinite dimensional (in physics)
it's injective because [tex]\Phi[/tex] is dense in H, you can prove injectivity in a few steps.
 

1. What is dual space topology?

Dual space topology is a mathematical concept that involves the study of topological spaces in which the elements are functions on another topological space. It is closely related to the concept of duality in mathematics, where one mathematical structure can be described in terms of another structure.

2. How is dual space topology different from traditional topology?

Dual space topology differs from traditional topology in that it focuses on the duality between two topological spaces, rather than just one. This allows for a more flexible and powerful way of studying topological spaces, as it takes into account both the space itself and the functions defined on it.

3. What is the purpose of the A to B inclusion map in dual space topology?

The A to B inclusion map is a key component in dual space topology, as it defines the relationship between two topological spaces. It maps elements from one space (A) to elements in another space (B), allowing for the comparison and analysis of the two spaces.

4. How is dual space topology used in real-world applications?

Dual space topology has a wide range of applications in various fields, including physics, engineering, and computer science. It is used to model and analyze complex systems, such as networks, circuits, and signal processing. It also has applications in data analysis and machine learning.

5. What are the benefits of studying dual space topology?

Studying dual space topology can lead to a deeper understanding of mathematical concepts and their applications. It also provides a powerful tool for analyzing and solving complex problems in various fields. Additionally, it can help bridge the gap between different areas of mathematics by revealing connections and duality between seemingly unrelated structures.

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