Function spaces from a geometrical viewpoint.
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I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in whether there are some geometrical difference between each functional space or TVS (local/global convexness? Alexandrov curvature?)(I found vector space is always global convex, sorry! in 11/Dec '18).
And I'd like to take a look at some references about this kind of topic.
Anything will help, thank you!
geometry functional-analysis reference-request topological-vector-spaces
asked Dec 9 '18 at 2:38
KeiKei
347
$endgroup$
add a comment |
$begingroup$
I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in whether there are some geometrical difference between each functional space or TVS (local/global convexness? Alexandrov curvature?)(I found vector space is always global convex, sorry! in 11/Dec '18).
And I'd like to take a look at some references about this kind of topic.
Anything will help, thank you!
geometry functional-analysis reference-request topological-vector-spaces
asked Dec 9 '18 at 2:38
KeiKei
347
$endgroup$
$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
1
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@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
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No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
1
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
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– Giuseppe Negro
Dec 11 '18 at 11:04
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@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25
add a comment |
$begingroup$
I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in whether there are some geometrical difference between each functional space or TVS (local/global convexness? Alexandrov curvature?)(I found vector space is always global convex, sorry! in 11/Dec '18).
And I'd like to take a look at some references about this kind of topic.
Anything will help, thank you!
geometry functional-analysis reference-request topological-vector-spaces
asked Dec 9 '18 at 2:38
KeiKei
347
$endgroup$
I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in whether there are some geometrical difference between each functional space or TVS (local/global convexness? Alexandrov curvature?)(I found vector space is always global convex, sorry! in 11/Dec '18).
And I'd like to take a look at some references about this kind of topic.
Anything will help, thank you!
geometry functional-analysis reference-request topological-vector-spaces
geometry functional-analysis reference-request topological-vector-spaces
asked Dec 9 '18 at 2:38
KeiKei
347
asked Dec 9 '18 at 2:38
KeiKei
347
edited Dec 11 '18 at 11:01
Kei
asked Dec 9 '18 at 2:38
KeiKei
347
asked Dec 9 '18 at 2:38
KeiKei
347
asked Dec 9 '18 at 2:38
KeiKei
347
347
$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
1
$begingroup$
@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
$begingroup$
No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
1
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
$endgroup$
– Giuseppe Negro
Dec 11 '18 at 11:04
$begingroup$
@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25
add a comment |
$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
1
$begingroup$
@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
$begingroup$
No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
1
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
$endgroup$
– Giuseppe Negro
Dec 11 '18 at 11:04
$begingroup$
@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25
$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
1
1
$begingroup$
@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
$begingroup$
@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
$begingroup$
No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
$begingroup$
No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
1
1
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
$endgroup$
– Giuseppe Negro
Dec 11 '18 at 11:04
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
$endgroup$
– Giuseppe Negro
Dec 11 '18 at 11:04
$begingroup$
@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25
$begingroup$
@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25
add a comment |
1 Answer
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oldest
votes
$begingroup$
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1le p,q le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,qge 1$, if $max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
$endgroup$
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
add a comment |
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1 Answer
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$begingroup$
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1le p,q le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,qge 1$, if $max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
$endgroup$
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
add a comment |
$begingroup$
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1le p,q le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,qge 1$, if $max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
$endgroup$
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
add a comment |
$begingroup$
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1le p,q le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,qge 1$, if $max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
$endgroup$
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1le p,q le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,qge 1$, if $max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
answered Dec 9 '18 at 3:46
BigbearZzzBigbearZzz
8,58221652
8,58221652
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
add a comment |
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
$begingroup$
Thanks! I found "nonexistence of uniform homeomorphisms between $L^p$ spaces" interesting. Is "uniformity" condition important in this kind of theory?
$endgroup$
– Kei
Dec 9 '18 at 3:58
add a comment |
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$begingroup$
There's no real need to include "reference request" in the title if you use the tag of the same name.
$endgroup$
– Shaun
Dec 9 '18 at 2:41
1
$begingroup$
@Shan I'm sorry and thanks for your edit.
$endgroup$
– Kei
Dec 9 '18 at 2:54
$begingroup$
No apologies necessary. You're welcome :)
$endgroup$
– Shaun
Dec 9 '18 at 2:56
1
$begingroup$
This is a whole branch of mathematics, it is called "Geometry of Banach spaces". It is a vast subject and it is impossible to resume it in an answer on Math.SE. If you are curious, you can consult some books on the subject; there are many. A standard reference is Lindenstrauss-Tzafriri.
$endgroup$
– Giuseppe Negro
Dec 11 '18 at 11:04
$begingroup$
@GiuseppeNegro Thank you! I'd like to know, what kind of words or phrases titles may have for this kind of topic? (or this question is also too vast?)
$endgroup$
– Kei
Dec 11 '18 at 11:25