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Why are uniform spaces important?
The Next CEO of Stack OverflowTopology on the tensor product of two topological vector spaces — how properties does it maintains?Intuition behind the failure of unimodularityGeneralization of inner product spaces (analogue to uniform spaces/locally convex spaces)Basic questions about lattices, fundamental domains, and quotient spacesUnderstanding Wikipedia's definition for latticeWhat is the purpose of this axiom in the definition of a uniform space?How is the modular character usually defined?Completeness of Continuous Functions on Uniform SpacesWhy are compact Hausdorff spaces divisible - and other uniformity related questionsLimit point compact uniform space
$begingroup$
Why are uniform spaces important? I've thought of two possible good answers:
Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and the Haar measure is defined on locally compact groups.
Functional analysis. Every seminorm induces a pseudometric, and I've read that uniform spaces can be defined via pseudometrics. This suggests to me that important results like the Hahn-Banach theorem would hold for complete uniform spaces. Is this true?
Unfortunately I lack the background to understand the Haar measure properly, and I'm only familiar with the entourage version of uniform spaces.
Any exposition and thoughts on what I've written above would be highly appreciated! Other important consequences of uniform structures would be great too.
general-topology functional-analysis soft-question topological-groups uniform-spaces
edited yesterday
YuiTo Cheng
2,1412937
asked yesterday
jessicajessica
7719
$endgroup$
add a comment |
$begingroup$
Why are uniform spaces important? I've thought of two possible good answers:
Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and the Haar measure is defined on locally compact groups.
Functional analysis. Every seminorm induces a pseudometric, and I've read that uniform spaces can be defined via pseudometrics. This suggests to me that important results like the Hahn-Banach theorem would hold for complete uniform spaces. Is this true?
Unfortunately I lack the background to understand the Haar measure properly, and I'm only familiar with the entourage version of uniform spaces.
Any exposition and thoughts on what I've written above would be highly appreciated! Other important consequences of uniform structures would be great too.
general-topology functional-analysis soft-question topological-groups uniform-spaces
edited yesterday
YuiTo Cheng
2,1412937
asked yesterday
jessicajessica
7719
$endgroup$
1
$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday
add a comment |
$begingroup$
Why are uniform spaces important? I've thought of two possible good answers:
Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and the Haar measure is defined on locally compact groups.
Functional analysis. Every seminorm induces a pseudometric, and I've read that uniform spaces can be defined via pseudometrics. This suggests to me that important results like the Hahn-Banach theorem would hold for complete uniform spaces. Is this true?
Unfortunately I lack the background to understand the Haar measure properly, and I'm only familiar with the entourage version of uniform spaces.
Any exposition and thoughts on what I've written above would be highly appreciated! Other important consequences of uniform structures would be great too.
general-topology functional-analysis soft-question topological-groups uniform-spaces
edited yesterday
YuiTo Cheng
2,1412937
asked yesterday
jessicajessica
7719
$endgroup$
Why are uniform spaces important? I've thought of two possible good answers:
Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and the Haar measure is defined on locally compact groups.
Functional analysis. Every seminorm induces a pseudometric, and I've read that uniform spaces can be defined via pseudometrics. This suggests to me that important results like the Hahn-Banach theorem would hold for complete uniform spaces. Is this true?
Unfortunately I lack the background to understand the Haar measure properly, and I'm only familiar with the entourage version of uniform spaces.
Any exposition and thoughts on what I've written above would be highly appreciated! Other important consequences of uniform structures would be great too.
general-topology functional-analysis soft-question topological-groups uniform-spaces
general-topology functional-analysis soft-question topological-groups uniform-spaces
edited yesterday
YuiTo Cheng
2,1412937
asked yesterday
jessicajessica
7719
edited yesterday
YuiTo Cheng
2,1412937
asked yesterday
jessicajessica
7719
edited yesterday
YuiTo Cheng
2,1412937
edited yesterday
YuiTo Cheng
2,1412937
edited yesterday
YuiTo Cheng
2,1412937
2,1412937
asked yesterday
jessicajessica
7719
asked yesterday
jessicajessica
7719
asked yesterday
jessicajessica
7719
7719
1
$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday
add a comment |
1
$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday
1
1
$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday
add a comment |
0
active
oldest
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$begingroup$
Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics
$endgroup$
– Max
yesterday
$begingroup$
Thanks for the suggestion. Do you have specific examples or theorems I should look at?
$endgroup$
– jessica
yesterday