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?






functional-analysis hilbert-spaces







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














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







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?






functional-analysis hilbert-spaces







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




functional-analysis hilbert-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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






active

oldest

votes


















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





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061489%2funcountable-sum-of-vectors-in-a-hilbert-space%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      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






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061489%2funcountable-sum-of-vectors-in-a-hilbert-space%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Understanding the information contained in the Deep Space Network XML data?

                      Image
                      up vote 1 down vote favorite NASA's Deep Space Network activity can be viewed conveniently at https://eyes.nasa.gov/dsn/dsn.html but if you would like data in a more raw XML form, it can be accessed as described in this answer Below is a small Python script that reads the XML into a list of dictionaries, and saves it to disk as a JSON. The dictionaries are organized at the higher level by dish, then by spacecraft per dish. Question: What are the most-likely meanings of all the data field contained in the Deep Space Network XML data? This is the example url from the other answer, used in the script below: https://eyes.nasa.gov/dsn/data/1365107113.xml def dictify(r,root=True): # https://stackoverflow.com/questions/2148119/how-to-convert-an-xml-string-to-a-dictionary-in-python/30923963#30923963 if root: ret
                      Read more

                      Ross-on-Wye

                      Image
                      Ross-on-Wye Welsh: Rhosan ar Wy Town centre, looking north from Market House Ross-on-Wye Ross-on-Wye shown within Herefordshire Population 10,700 (2011) [1] OS grid reference SO597241 Unitary authority Herefordshire Ceremonial county Herefordshire Region West Midlands Country England Sovereign state United Kingdom Post town ROSS-ON-WYE Postcode district HR9 Dialling code 01989 Police West Mercia Fire Hereford and Worcester Ambulance West Midlands EU Parliament West Midlands UK Parliament Hereford and South Herefordshire List of places UK England Herefordshire 51°54′50″N 2°35′13″W  /  51.914°N 2.587°W  / 51.914; -2.587 Coordinates: 51°54′50″N 2°35′13″W  /  51.914°N 2.587°W  / 51.914; -2.587 Ross-on-Wye (Welsh: Rhosan ar Wy [2] ) is a small market town with a population of 10,700 (according to the 2011 census), in south eastern Herefordshire, England, located on the River Wye, and on
                      Read more

                      Eastern Orthodox Church

                      Image
                      Not to be confused with Oriental Orthodoxy, Eastern Christianity, or Eastern Catholic Churches. "Eastern Orthodoxy", "Orthodox Church", "Orthodox Catholic Church", and "Orthodox Christian Church" redirect here. For other uses, see Orthodox (disambiguation). This article may be too long to read and navigate comfortably . The readable prose size is 132 kilobytes. Please consider splitting content into sub-articles, condensing it, or adding or removing subheadings. ( January 2018 ) Part of a series on the Eastern Orthodox Church Mosaic of Christ Pantocrator, Hagia Sophia Overview Structure Theology (History of theology) Liturgy Church history Holy Mysteries View of salvation View of Mary View of icons Background Crucifixion / Resurrection / Ascension of Jesus Christianity Christian Church Apostolic succession Four Marks of the Church Orthodoxy Organization Autocephaly Patriarchate
                      Read more