Uncountable sum of vectors in a Hilbert Space












2














I am currently reading Hilbert Spaces and confused about a thing. Say, $C={e_alpha : alphainmathcal{A}}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $sum_{alphainmathcal{A}}e_alpha$ well defined ? I think it should be, is there some kind of convergence needed for these sums?










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  • 2




    Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
    – Kavi Rama Murthy
    45 mins ago










  • Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
    – PSG
    42 mins ago






  • 1




    If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
    – Kavi Rama Murthy
    39 mins ago


















2














I am currently reading Hilbert Spaces and confused about a thing. Say, $C={e_alpha : alphainmathcal{A}}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $sum_{alphainmathcal{A}}e_alpha$ well defined ? I think it should be, is there some kind of convergence needed for these sums?










share|cite|improve this question


















  • 2




    Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
    – Kavi Rama Murthy
    45 mins ago










  • Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
    – PSG
    42 mins ago






  • 1




    If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
    – Kavi Rama Murthy
    39 mins ago
















2












2








2







I am currently reading Hilbert Spaces and confused about a thing. Say, $C={e_alpha : alphainmathcal{A}}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $sum_{alphainmathcal{A}}e_alpha$ well defined ? I think it should be, is there some kind of convergence needed for these sums?










share|cite|improve this question













I am currently reading Hilbert Spaces and confused about a thing. Say, $C={e_alpha : alphainmathcal{A}}$ be a complete orthonormal set of a Hilbert Space $H$, possibly uncountable. Is $sum_{alphainmathcal{A}}e_alpha$ well defined ? I think it should be, is there some kind of convergence needed for these sums?







functional-analysis hilbert-spaces






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asked 51 mins ago









PSG

3609




3609








  • 2




    Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
    – Kavi Rama Murthy
    45 mins ago










  • Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
    – PSG
    42 mins ago






  • 1




    If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
    – Kavi Rama Murthy
    39 mins ago
















  • 2




    Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
    – Kavi Rama Murthy
    45 mins ago










  • Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
    – PSG
    42 mins ago






  • 1




    If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
    – Kavi Rama Murthy
    39 mins ago










2




2




Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
– Kavi Rama Murthy
45 mins ago




Even when $mathcal A$ is countably infinite the sum $sum e_n$ does not exist either in the norm or weakly.
– Kavi Rama Murthy
45 mins ago












Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
– PSG
42 mins ago




Oh, I see if it was a well defined vector, then the norm would be in $mathbf{R}$, which is not the case here. Is that ok? @KaviRamaMurthy
– PSG
42 mins ago




1




1




If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
– Kavi Rama Murthy
39 mins ago






If $sum e_n=x$, say in weak topology, then we have $sum langle y,e_nrangle =langle y,xrangle$ for all $y$. You get a contradiction by taking $y=sum frac 1 j e_j$.
– Kavi Rama Murthy
39 mins ago












2 Answers
2






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4














In an infinite dimensional Hilbert space the term
$$
sum_{ain mathcal A} e_a
$$

is not well defined.
Even for countable $mathcal A$ this sum does not converge.



For defining uncountable sums it is usually required that at most countable many summands are nonzero
and that the countable sum over the nonzero entries converges absolutely.






share|cite|improve this answer





























    1














    The background of your question is how do you define $displaystyle sum_{i in I} c_i$ in a Banach space for any set $I$?



    The usual way is to say that the sum $displaystyle sum_{i in I} c_alpha$ of a set $mathcal C = {c_i ; i in I}$ of vectors exists when following Cauchy criteria is met:



    $$(forall epsilon > 0) , (exists J_0 in mathcal F(I)) , (forall K in mathcal F(I setminus J_0)) , leftVert displaystyle sum_{k in K} c_k rightVert< epsilon $$



    Where $mathcal F(A)$ is defined as the sets of finite subsets of $A$.



    This has interesting consequences.




    1. Such a sum $displaystyle sum_{i in I} c_alpha$ does make sense only if the set of non zero elements of $mathcal C$ is at most countable.

    2. For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $epsilon = 1/2$ in the definition above.






    share|cite|improve this answer





















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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

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      active

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      active

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      4














      In an infinite dimensional Hilbert space the term
      $$
      sum_{ain mathcal A} e_a
      $$

      is not well defined.
      Even for countable $mathcal A$ this sum does not converge.



      For defining uncountable sums it is usually required that at most countable many summands are nonzero
      and that the countable sum over the nonzero entries converges absolutely.






      share|cite|improve this answer


























        4














        In an infinite dimensional Hilbert space the term
        $$
        sum_{ain mathcal A} e_a
        $$

        is not well defined.
        Even for countable $mathcal A$ this sum does not converge.



        For defining uncountable sums it is usually required that at most countable many summands are nonzero
        and that the countable sum over the nonzero entries converges absolutely.






        share|cite|improve this answer
























          4












          4








          4






          In an infinite dimensional Hilbert space the term
          $$
          sum_{ain mathcal A} e_a
          $$

          is not well defined.
          Even for countable $mathcal A$ this sum does not converge.



          For defining uncountable sums it is usually required that at most countable many summands are nonzero
          and that the countable sum over the nonzero entries converges absolutely.






          share|cite|improve this answer












          In an infinite dimensional Hilbert space the term
          $$
          sum_{ain mathcal A} e_a
          $$

          is not well defined.
          Even for countable $mathcal A$ this sum does not converge.



          For defining uncountable sums it is usually required that at most countable many summands are nonzero
          and that the countable sum over the nonzero entries converges absolutely.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 43 mins ago









          supinf

          5,9991027




          5,9991027























              1














              The background of your question is how do you define $displaystyle sum_{i in I} c_i$ in a Banach space for any set $I$?



              The usual way is to say that the sum $displaystyle sum_{i in I} c_alpha$ of a set $mathcal C = {c_i ; i in I}$ of vectors exists when following Cauchy criteria is met:



              $$(forall epsilon > 0) , (exists J_0 in mathcal F(I)) , (forall K in mathcal F(I setminus J_0)) , leftVert displaystyle sum_{k in K} c_k rightVert< epsilon $$



              Where $mathcal F(A)$ is defined as the sets of finite subsets of $A$.



              This has interesting consequences.




              1. Such a sum $displaystyle sum_{i in I} c_alpha$ does make sense only if the set of non zero elements of $mathcal C$ is at most countable.

              2. For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $epsilon = 1/2$ in the definition above.






              share|cite|improve this answer


























                1














                The background of your question is how do you define $displaystyle sum_{i in I} c_i$ in a Banach space for any set $I$?



                The usual way is to say that the sum $displaystyle sum_{i in I} c_alpha$ of a set $mathcal C = {c_i ; i in I}$ of vectors exists when following Cauchy criteria is met:



                $$(forall epsilon > 0) , (exists J_0 in mathcal F(I)) , (forall K in mathcal F(I setminus J_0)) , leftVert displaystyle sum_{k in K} c_k rightVert< epsilon $$



                Where $mathcal F(A)$ is defined as the sets of finite subsets of $A$.



                This has interesting consequences.




                1. Such a sum $displaystyle sum_{i in I} c_alpha$ does make sense only if the set of non zero elements of $mathcal C$ is at most countable.

                2. For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $epsilon = 1/2$ in the definition above.






                share|cite|improve this answer
























                  1












                  1








                  1






                  The background of your question is how do you define $displaystyle sum_{i in I} c_i$ in a Banach space for any set $I$?



                  The usual way is to say that the sum $displaystyle sum_{i in I} c_alpha$ of a set $mathcal C = {c_i ; i in I}$ of vectors exists when following Cauchy criteria is met:



                  $$(forall epsilon > 0) , (exists J_0 in mathcal F(I)) , (forall K in mathcal F(I setminus J_0)) , leftVert displaystyle sum_{k in K} c_k rightVert< epsilon $$



                  Where $mathcal F(A)$ is defined as the sets of finite subsets of $A$.



                  This has interesting consequences.




                  1. Such a sum $displaystyle sum_{i in I} c_alpha$ does make sense only if the set of non zero elements of $mathcal C$ is at most countable.

                  2. For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $epsilon = 1/2$ in the definition above.






                  share|cite|improve this answer












                  The background of your question is how do you define $displaystyle sum_{i in I} c_i$ in a Banach space for any set $I$?



                  The usual way is to say that the sum $displaystyle sum_{i in I} c_alpha$ of a set $mathcal C = {c_i ; i in I}$ of vectors exists when following Cauchy criteria is met:



                  $$(forall epsilon > 0) , (exists J_0 in mathcal F(I)) , (forall K in mathcal F(I setminus J_0)) , leftVert displaystyle sum_{k in K} c_k rightVert< epsilon $$



                  Where $mathcal F(A)$ is defined as the sets of finite subsets of $A$.



                  This has interesting consequences.




                  1. Such a sum $displaystyle sum_{i in I} c_alpha$ does make sense only if the set of non zero elements of $mathcal C$ is at most countable.

                  2. For a family of uncountable vectors of norm equal to $1$, which is your initial question, the sum cannot exist: take $epsilon = 1/2$ in the definition above.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 15 mins ago









                  mathcounterexamples.net

                  24.8k21753




                  24.8k21753






























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