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Does a sequence of distribution converges if it converges on a dense subset of test function space?

Solving the equation $nabla u=f$.Does this distribution make any sense?Extension of a Pseudodifferential OperatorShow that a particular sum converges and defines a distribution (example from Friedlander)A sequence of distributions converges to a certain distribution.Show that continuous linear maps on the space of test functions take $C_K^infty(Omega)$ into some $C_K_N^infty(Omega)$If two distributions vanish on the same set of test functions, then one is a constant multiple of the otherWhy does convolution of delta function commute (test distribution perspective)?Why test space is a subset of Schwartz space?Proof related to tensor of distributions properties

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1. Suppose $K_n subset mathcalD'$, and $langle K_n, urangle$ converges for every $u$ in a dense subset of $C_0^infty$, does $K_n$ converges to some $Kin mathcalD'$?

If this is not true, how about:

2. Suppose $K_n subset mathcalD', K in mathcalD'$, and $langle K_n, urangle to langle K, urangle$, for every $u$ in a dense subset of $C^infty_0$, does $K_n$ converges to $K$?

A proof or a counterexample would be appreciated.

distribution-theory

share|cite|improve this question

asked Mar 18 at 0:01

user655213user655213

161

$endgroup$

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  I would start by trying to prove that $langle K_n,urangle$ is a Cauchy sequence for any $uin C_0^infty$. It is natural to approximate $u$ by some $u_k$ in your dense set, but then you need some sort of uniformity to be able to take limits simultaneously in $n$ and $k$. Question: what do you mean by dense in $C_0^infty$? Do you mean in the sense of the sup. norm of derivatives? Is any assumption on the convergence of the supports made? I am saying this because convergence in $mathcalD$ involves a uniform control on the support and is thus stronger than convergence in $C_0^infty$..
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  – GReyes
  Mar 18 at 1:18
  

  
  
  
  


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  @GReyes It is dense with respect to the countable family of semi-norms $sup_x _alpha, N$.
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  – user655213
  Mar 18 at 3:21
  
  

  
  
  
  


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  @GReyes what kind of uniformity can we use here? I'm not familiar with the techniques in distribution theory. So far I have tried using the Banach-Steinhaus theorem (in Frechet space) to show equicontinuity of $K_n$: for every $uin C_0^infty$, take a sequence $u_k$ in the dense set convergent to $u$, for every $epsilon >0$, there exists $N$, for all $kgeq N$, and for all $n$, $|langle K_n, u_krangle|<epsilon$. This uniformity would work. However, since we only have boundedness on a subset of $C_0^infty$, which is not a Frechet space, cannot apply Banach-Steinhaus theorem.
  $endgroup$
  – user655213
  Mar 18 at 3:32
  
  

  
  
  
  

add a comment | 

1

2

$begingroup$

1. Suppose $K_n subset mathcalD'$, and $langle K_n, urangle$ converges for every $u$ in a dense subset of $C_0^infty$, does $K_n$ converges to some $Kin mathcalD'$?

If this is not true, how about:

2. Suppose $K_n subset mathcalD', K in mathcalD'$, and $langle K_n, urangle to langle K, urangle$, for every $u$ in a dense subset of $C^infty_0$, does $K_n$ converges to $K$?

A proof or a counterexample would be appreciated.

distribution-theory

share|cite|improve this question

asked Mar 18 at 0:01

user655213user655213

161

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I would start by trying to prove that $langle K_n,urangle$ is a Cauchy sequence for any $uin C_0^infty$. It is natural to approximate $u$ by some $u_k$ in your dense set, but then you need some sort of uniformity to be able to take limits simultaneously in $n$ and $k$. Question: what do you mean by dense in $C_0^infty$? Do you mean in the sense of the sup. norm of derivatives? Is any assumption on the convergence of the supports made? I am saying this because convergence in $mathcalD$ involves a uniform control on the support and is thus stronger than convergence in $C_0^infty$..
  $endgroup$
  – GReyes
  Mar 18 at 1:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @GReyes It is dense with respect to the countable family of semi-norms $sup_x _alpha, N$.
  $endgroup$
  – user655213
  Mar 18 at 3:21
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @GReyes what kind of uniformity can we use here? I'm not familiar with the techniques in distribution theory. So far I have tried using the Banach-Steinhaus theorem (in Frechet space) to show equicontinuity of $K_n$: for every $uin C_0^infty$, take a sequence $u_k$ in the dense set convergent to $u$, for every $epsilon >0$, there exists $N$, for all $kgeq N$, and for all $n$, $|langle K_n, u_krangle|<epsilon$. This uniformity would work. However, since we only have boundedness on a subset of $C_0^infty$, which is not a Frechet space, cannot apply Banach-Steinhaus theorem.
  $endgroup$
  – user655213
  Mar 18 at 3:32
  
  

  
  
  
  

add a comment | 

$begingroup$

1. Suppose $K_n subset mathcalD'$, and $langle K_n, urangle$ converges for every $u$ in a dense subset of $C_0^infty$, does $K_n$ converges to some $Kin mathcalD'$?

If this is not true, how about:

2. Suppose $K_n subset mathcalD', K in mathcalD'$, and $langle K_n, urangle to langle K, urangle$, for every $u$ in a dense subset of $C^infty_0$, does $K_n$ converges to $K$?

A proof or a counterexample would be appreciated.

distribution-theory

share|cite|improve this question

asked Mar 18 at 0:01

user655213user655213

161

$endgroup$

share|cite|improve this question

asked Mar 18 at 0:01

user655213user655213

161

share|cite|improve this question

asked Mar 18 at 0:01

user655213user655213

161

asked Mar 18 at 0:01

user655213user655213

161

asked Mar 18 at 0:01

user655213user655213

161