How to define norm on semi vector spaces?

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And and can we give proofs different from vector space norm proofs with its restriction?Can we give standard proofs with the lack of additive identity and additive inverse property?

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asked Jul 28 at 15:31

Sherlin

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And and can we give proofs different from vector space norm proofs with its restriction?Can we give standard proofs with the lack of additive identity and additive inverse property?

functional-analysis

share|cite|improve this question

asked Jul 28 at 15:31

Sherlin

13

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

And and can we give proofs different from vector space norm proofs with its restriction?Can we give standard proofs with the lack of additive identity and additive inverse property?

functional-analysis

share|cite|improve this question

asked Jul 28 at 15:31

Sherlin

13

And and can we give proofs different from vector space norm proofs with its restriction?Can we give standard proofs with the lack of additive identity and additive inverse property?

functional-analysis

share|cite|improve this question

asked Jul 28 at 15:31

Sherlin

13

share|cite|improve this question

share|cite|improve this question

asked Jul 28 at 15:31

Sherlin

13

asked Jul 28 at 15:31

Sherlin

13

asked Jul 28 at 15:31

Sherlin

13

13

add a comment | 

add a comment | 

1 Answer
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No in general. For instance, the gliding hump proof of the uniform boundedness principle fails horribly. Also, the Hahn-Banach theorem can't be proven in the usual way because we can't pass to zero or to a subspace.

The strategy to obtain results would be to embed the semi vector space into some vector space. This works e.g. when the semi vector space has some sort of "basis".

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answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thnx can u give me some more properties..
  – Sherlin
  Jul 28 at 15:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would help greatly if you had a particular semi vector space in mind...
  – AlgebraicsAnonymous
  Jul 28 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
  – AlgebraicsAnonymous
  Jul 28 at 15:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
  – Sherlin
  Jul 28 at 15:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
  – AlgebraicsAnonymous
  Jul 28 at 15:52
  

  
  
  
  

add a comment | 

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1 Answer
1

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1 Answer
1

active

oldest

votes

active

oldest

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active

oldest

votes

up vote
0
down vote

No in general. For instance, the gliding hump proof of the uniform boundedness principle fails horribly. Also, the Hahn-Banach theorem can't be proven in the usual way because we can't pass to zero or to a subspace.

The strategy to obtain results would be to embed the semi vector space into some vector space. This works e.g. when the semi vector space has some sort of "basis".

share|cite|improve this answer

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thnx can u give me some more properties..
  – Sherlin
  Jul 28 at 15:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would help greatly if you had a particular semi vector space in mind...
  – AlgebraicsAnonymous
  Jul 28 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
  – AlgebraicsAnonymous
  Jul 28 at 15:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
  – Sherlin
  Jul 28 at 15:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
  – AlgebraicsAnonymous
  Jul 28 at 15:52
  

  
  
  
  

add a comment | 

up vote
0
down vote

No in general. For instance, the gliding hump proof of the uniform boundedness principle fails horribly. Also, the Hahn-Banach theorem can't be proven in the usual way because we can't pass to zero or to a subspace.

The strategy to obtain results would be to embed the semi vector space into some vector space. This works e.g. when the semi vector space has some sort of "basis".

share|cite|improve this answer

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thnx can u give me some more properties..
  – Sherlin
  Jul 28 at 15:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would help greatly if you had a particular semi vector space in mind...
  – AlgebraicsAnonymous
  Jul 28 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
  – AlgebraicsAnonymous
  Jul 28 at 15:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
  – Sherlin
  Jul 28 at 15:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
  – AlgebraicsAnonymous
  Jul 28 at 15:52
  

  
  
  
  

add a comment | 

up vote
0
down vote

up vote
0
down vote

No in general. For instance, the gliding hump proof of the uniform boundedness principle fails horribly. Also, the Hahn-Banach theorem can't be proven in the usual way because we can't pass to zero or to a subspace.

The strategy to obtain results would be to embed the semi vector space into some vector space. This works e.g. when the semi vector space has some sort of "basis".

share|cite|improve this answer

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

No in general. For instance, the gliding hump proof of the uniform boundedness principle fails horribly. Also, the Hahn-Banach theorem can't be proven in the usual way because we can't pass to zero or to a subspace.

The strategy to obtain results would be to embed the semi vector space into some vector space. This works e.g. when the semi vector space has some sort of "basis".

share|cite|improve this answer

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

answered Jul 28 at 15:36

AlgebraicsAnonymous

66611

66611

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thnx can u give me some more properties..
  – Sherlin
  Jul 28 at 15:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would help greatly if you had a particular semi vector space in mind...
  – AlgebraicsAnonymous
  Jul 28 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
  – AlgebraicsAnonymous
  Jul 28 at 15:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
  – Sherlin
  Jul 28 at 15:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
  – AlgebraicsAnonymous
  Jul 28 at 15:52
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thnx can u give me some more properties..
  – Sherlin
  Jul 28 at 15:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It would help greatly if you had a particular semi vector space in mind...
  – AlgebraicsAnonymous
  Jul 28 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
  – AlgebraicsAnonymous
  Jul 28 at 15:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
  – Sherlin
  Jul 28 at 15:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
  – AlgebraicsAnonymous
  Jul 28 at 15:52
  

  
  
  
  

Thnx can u give me some more properties..
– Sherlin
Jul 28 at 15:40

Thnx can u give me some more properties..
– Sherlin
Jul 28 at 15:40

It would help greatly if you had a particular semi vector space in mind...
– AlgebraicsAnonymous
Jul 28 at 15:41

It would help greatly if you had a particular semi vector space in mind...
– AlgebraicsAnonymous
Jul 28 at 15:41

If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
– AlgebraicsAnonymous
Jul 28 at 15:43

If you, for instance, were to talk about a convex cone in a Hilbert space, then things would work fine...
– AlgebraicsAnonymous
Jul 28 at 15:43

Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
– Sherlin
Jul 28 at 15:50

Can it work with 0 included and defining inner product on it and can we get different proofs ..i am specific abt the semi field of positive real numbers can u give me some simple semi vector spaces so that i can try to bring its norm properties
– Sherlin
Jul 28 at 15:50

Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
– AlgebraicsAnonymous
Jul 28 at 15:52

Possibly one could formally define the extension to the real numbers as a positive and negative part, just like the Grothendieck group construction. In this case, one would have an embedding and could get the results one wants from that.
– AlgebraicsAnonymous
Jul 28 at 15:52

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