If $(a_n)_nsubseteq Bbb{R}$ and $leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n)$, then $Tin...












0












$begingroup$



Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$




My trial



It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.



$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?










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$endgroup$












  • $begingroup$
    The only $a_n$ that matter are the ones for $nleq a$
    $endgroup$
    – Max
    Feb 1 at 14:06










  • $begingroup$
    @Max: The ones such that $nleq a$? How?
    $endgroup$
    – Micheal
    Feb 1 at 14:08






  • 1




    $begingroup$
    $displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
    $endgroup$
    – Max
    Feb 1 at 14:09










  • $begingroup$
    @Max: You made a point.
    $endgroup$
    – Micheal
    Feb 1 at 14:13
















0












$begingroup$



Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$




My trial



It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.



$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?










share|cite|improve this question











$endgroup$












  • $begingroup$
    The only $a_n$ that matter are the ones for $nleq a$
    $endgroup$
    – Max
    Feb 1 at 14:06










  • $begingroup$
    @Max: The ones such that $nleq a$? How?
    $endgroup$
    – Micheal
    Feb 1 at 14:08






  • 1




    $begingroup$
    $displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
    $endgroup$
    – Max
    Feb 1 at 14:09










  • $begingroup$
    @Max: You made a point.
    $endgroup$
    – Micheal
    Feb 1 at 14:13














0












0








0


1



$begingroup$



Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$




My trial



It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.



$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?










share|cite|improve this question











$endgroup$





Let $(a_n)_nsubseteq Bbb{R}$ such that $$leftlangle T,varphi rightrangle=sum^{infty}_{n=0}a_nvarphi(n).$$
I want to prove that $Tin D'(Bbb{R}).$




My trial



It suffices to prove that $Tin l^1_{loc}(Bbb{R})$. That is,
$$ sum^{infty}_{n=0}left|a_nvarphi(n)right|<infty.$$
Let $varphiin D(Bbb{R})$, then there exists $a>0$ such that $text{supp}varphisubseteq [-a,a]$, where supp is the support of $varphi$.



$$left|leftlangle T,varphi rightrangleright|leq sum^{infty}_{n=0}left|a_nright| left|varphi(n)right|leq sup_{nin [-a,a]} left|varphi(n)right|sum^{infty}_{n=0}left|a_nright|.$$
Since $(a_n)_nsubseteq Bbb{R}$, I am not sure that $sum^{infty}_{n=0}left|a_nright|<infty.$ So, I am stuck here, as I don't know how to proceed. Any help, please?







functional-analysis distribution-theory






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share|cite|improve this question













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edited Feb 1 at 14:00







Micheal

















asked Feb 1 at 13:48









MichealMicheal

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26511












  • $begingroup$
    The only $a_n$ that matter are the ones for $nleq a$
    $endgroup$
    – Max
    Feb 1 at 14:06










  • $begingroup$
    @Max: The ones such that $nleq a$? How?
    $endgroup$
    – Micheal
    Feb 1 at 14:08






  • 1




    $begingroup$
    $displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
    $endgroup$
    – Max
    Feb 1 at 14:09










  • $begingroup$
    @Max: You made a point.
    $endgroup$
    – Micheal
    Feb 1 at 14:13


















  • $begingroup$
    The only $a_n$ that matter are the ones for $nleq a$
    $endgroup$
    – Max
    Feb 1 at 14:06










  • $begingroup$
    @Max: The ones such that $nleq a$? How?
    $endgroup$
    – Micheal
    Feb 1 at 14:08






  • 1




    $begingroup$
    $displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
    $endgroup$
    – Max
    Feb 1 at 14:09










  • $begingroup$
    @Max: You made a point.
    $endgroup$
    – Micheal
    Feb 1 at 14:13
















$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06




$begingroup$
The only $a_n$ that matter are the ones for $nleq a$
$endgroup$
– Max
Feb 1 at 14:06












$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08




$begingroup$
@Max: The ones such that $nleq a$? How?
$endgroup$
– Micheal
Feb 1 at 14:08




1




1




$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09




$begingroup$
$displaystyle|sum_{n=0}^infty a_nvarphi(n)| = |sum_{nleq a} a_nvarphi(n) + 0| leq ||varphi||_K displaystylesum_{nleq a } |a_n|$
$endgroup$
– Max
Feb 1 at 14:09












$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13




$begingroup$
@Max: You made a point.
$endgroup$
– Micheal
Feb 1 at 14:13










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