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Convergence of scalar product in a hilbert space

0

$begingroup$

Background: From this paper I'm trying to understand why the OPE in conformal field theory has a finite radius of convergence.

The authors make the claim that

the scalar product of two states converges when one of the two states is expanded into an
orthonormal basis.

Here a "state" is a ray in a hilbert space.

1. Does the following argument prove the claim above? (Notation from Rudin chapter 4)

Let $phi, psi in ell^2(A) = H$ with $psi$ expressible in an orthonormal basis of $H$. Without loss of generality, assume $psi$ is normalized and one of the basis vectors of the basis mentioned. Then

$$(phi, psi) = sum_{ain A}phi(a)overline{psi}(a) = phi(0)$$

where by assumption $psi(a) = 1$ for $a = 0$ and vanishes elsewhere. Then since $phi in H$ the inner product converges.

2. Can it be shown that if neither state may be expanded in an orthonormal basis of $H$ then the scalar product may not converge?

Thanks!

functional-analysis hilbert-spaces mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You (or the paper) must be working with different definitions than usual. On a Hilbert space the scalar product of any two vectors is well defined, and any vector is expressible with respect to any orthonormal basis. At this point it would be helpful to be explicit about what you are working with.
  $endgroup$
  – s.harp
  Mar 22 at 23:03
  

  
  
  
  

add a comment | 

0

$begingroup$

Background: From this paper I'm trying to understand why the OPE in conformal field theory has a finite radius of convergence.

The authors make the claim that

the scalar product of two states converges when one of the two states is expanded into an
orthonormal basis.

Here a "state" is a ray in a hilbert space.

1. Does the following argument prove the claim above? (Notation from Rudin chapter 4)

Let $phi, psi in ell^2(A) = H$ with $psi$ expressible in an orthonormal basis of $H$. Without loss of generality, assume $psi$ is normalized and one of the basis vectors of the basis mentioned. Then

$$(phi, psi) = sum_{ain A}phi(a)overline{psi}(a) = phi(0)$$

where by assumption $psi(a) = 1$ for $a = 0$ and vanishes elsewhere. Then since $phi in H$ the inner product converges.

2. Can it be shown that if neither state may be expanded in an orthonormal basis of $H$ then the scalar product may not converge?

Thanks!

functional-analysis hilbert-spaces mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You (or the paper) must be working with different definitions than usual. On a Hilbert space the scalar product of any two vectors is well defined, and any vector is expressible with respect to any orthonormal basis. At this point it would be helpful to be explicit about what you are working with.
  $endgroup$
  – s.harp
  Mar 22 at 23:03
  

  
  
  
  

add a comment | 

$begingroup$

Background: From this paper I'm trying to understand why the OPE in conformal field theory has a finite radius of convergence.

The authors make the claim that

the scalar product of two states converges when one of the two states is expanded into an
orthonormal basis.

Here a "state" is a ray in a hilbert space.

1. Does the following argument prove the claim above? (Notation from Rudin chapter 4)

Let $phi, psi in ell^2(A) = H$ with $psi$ expressible in an orthonormal basis of $H$. Without loss of generality, assume $psi$ is normalized and one of the basis vectors of the basis mentioned. Then

$$(phi, psi) = sum_{ain A}phi(a)overline{psi}(a) = phi(0)$$

where by assumption $psi(a) = 1$ for $a = 0$ and vanishes elsewhere. Then since $phi in H$ the inner product converges.

2. Can it be shown that if neither state may be expanded in an orthonormal basis of $H$ then the scalar product may not converge?

Thanks!

functional-analysis hilbert-spaces mathematical-physics conformal-field-theory

share|cite|improve this question

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

$endgroup$

share|cite|improve this question

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

share|cite|improve this question

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

edited Mar 24 at 0:41

Andrews

1,2962423

edited Mar 24 at 0:41

Andrews

1,2962423

asked Mar 22 at 20:19

DiffycueDiffycue

401215

asked Mar 22 at 20:19

DiffycueDiffycue

401215