Closed Convex Hull












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Let $X$ be a topological vector space. For a countable subset subset $(a_n)_nsubset X$, can one describe the closed convex hull of $(a_n)_n$ as the set ${sum_{n=1}^{infty}c_na_n: 0leq c_nleq 1, text{and} sum_{n=1}^{infty}c_n=1}$ where the series converges?






functional-analysis







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    $begingroup$
    Possibly related: math.stackexchange.com/q/1915689/42969.
    $endgroup$
    – Martin R
    Feb 3 at 7:03










  • $begingroup$
    @MartinR In your counter example, I think the convex set is not (norm) closed.
    $endgroup$
    – user124910
    Feb 3 at 7:48
















2












$begingroup$


Let $X$ be a topological vector space. For a countable subset subset $(a_n)_nsubset X$, can one describe the closed convex hull of $(a_n)_n$ as the set ${sum_{n=1}^{infty}c_na_n: 0leq c_nleq 1, text{and} sum_{n=1}^{infty}c_n=1}$ where the series converges?






functional-analysis







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Possibly related: math.stackexchange.com/q/1915689/42969.
    $endgroup$
    – Martin R
    Feb 3 at 7:03










  • $begingroup$
    @MartinR In your counter example, I think the convex set is not (norm) closed.
    $endgroup$
    – user124910
    Feb 3 at 7:48














2












2








2





$begingroup$


Let $X$ be a topological vector space. For a countable subset subset $(a_n)_nsubset X$, can one describe the closed convex hull of $(a_n)_n$ as the set ${sum_{n=1}^{infty}c_na_n: 0leq c_nleq 1, text{and} sum_{n=1}^{infty}c_n=1}$ where the series converges?






functional-analysis







share|cite|improve this question









$endgroup$




Let $X$ be a topological vector space. For a countable subset subset $(a_n)_nsubset X$, can one describe the closed convex hull of $(a_n)_n$ as the set ${sum_{n=1}^{infty}c_na_n: 0leq c_nleq 1, text{and} sum_{n=1}^{infty}c_n=1}$ where the series converges?






functional-analysis




functional-analysis






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asked Feb 3 at 6:58









user124910user124910

1,257817




1,257817








  • 1




    $begingroup$
    Possibly related: math.stackexchange.com/q/1915689/42969.
    $endgroup$
    – Martin R
    Feb 3 at 7:03










  • $begingroup$
    @MartinR In your counter example, I think the convex set is not (norm) closed.
    $endgroup$
    – user124910
    Feb 3 at 7:48














  • 1




    $begingroup$
    Possibly related: math.stackexchange.com/q/1915689/42969.
    $endgroup$
    – Martin R
    Feb 3 at 7:03










  • $begingroup$
    @MartinR In your counter example, I think the convex set is not (norm) closed.
    $endgroup$
    – user124910
    Feb 3 at 7:48








1




1




$begingroup$
Possibly related: math.stackexchange.com/q/1915689/42969.
$endgroup$
– Martin R
Feb 3 at 7:03




$begingroup$
Possibly related: math.stackexchange.com/q/1915689/42969.
$endgroup$
– Martin R
Feb 3 at 7:03












$begingroup$
@MartinR In your counter example, I think the convex set is not (norm) closed.
$endgroup$
– user124910
Feb 3 at 7:48




$begingroup$
@MartinR In your counter example, I think the convex set is not (norm) closed.
$endgroup$
– user124910
Feb 3 at 7:48










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No.



Let $X=Bbb R$ and $a_n=1/n$ for $nge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0nesum c_na_n$ with $c_nge0$, $sum c_n=1$.






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    $begingroup$

    No.



    Let $X=Bbb R$ and $a_n=1/n$ for $nge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0nesum c_na_n$ with $c_nge0$, $sum c_n=1$.






    share|cite|improve this answer









    $endgroup$


















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      $begingroup$

      No.



      Let $X=Bbb R$ and $a_n=1/n$ for $nge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0nesum c_na_n$ with $c_nge0$, $sum c_n=1$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        No.



        Let $X=Bbb R$ and $a_n=1/n$ for $nge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0nesum c_na_n$ with $c_nge0$, $sum c_n=1$.






        share|cite|improve this answer









        $endgroup$



        No.



        Let $X=Bbb R$ and $a_n=1/n$ for $nge1$. The closed convex hull of the $a_n$ is $[0,1]$, but it's clear that $0nesum c_na_n$ with $c_nge0$, $sum c_n=1$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 3 at 14:25









        David C. UllrichDavid C. Ullrich

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