why a subspace is closed?

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Let $E$ be $K-$vector space with a norm $|cdot|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set .



I am trying to show that any convergent sequence of elements $(x_n)_n in N$ of $F$ converge to an element $x in F$.



This means I need to show that $||x_n-x|| rightarrow 0$



I see that like a distance which is attended by the fuction $y rightarrow ||x-y||$ .



How can I show that $ x in F$?



I have been reading a solution which I don't really understand the intuition behind it :



let $x$ be in $E$ and let $B=$
after showing that its a compact, they tried to show that $inf_y in F(||x-y||)= lambda $ exists.



by definition of $lambda$ it is the distance between $x$ and $F$










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  • 1




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    I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
    $endgroup$
    – K.Power
    Mar 2 at 0:44











  • $begingroup$
    @K.Power actually I am looking for both
    $endgroup$
    – user515918
    Mar 2 at 0:46










  • $begingroup$
    If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
    $endgroup$
    – K.Power
    Mar 2 at 0:47










  • $begingroup$
    @K.Power I've been searching but I dont have any idea about the methode they using
    $endgroup$
    – user515918
    Mar 2 at 0:48










  • $begingroup$
    As these are very distinct questions you should definitely ask these in two separate posts.
    $endgroup$
    – K.Power
    Mar 2 at 0:50















4












$begingroup$


Let $E$ be $K-$vector space with a norm $|cdot|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set .



I am trying to show that any convergent sequence of elements $(x_n)_n in N$ of $F$ converge to an element $x in F$.



This means I need to show that $||x_n-x|| rightarrow 0$



I see that like a distance which is attended by the fuction $y rightarrow ||x-y||$ .



How can I show that $ x in F$?



I have been reading a solution which I don't really understand the intuition behind it :



let $x$ be in $E$ and let $B=$
after showing that its a compact, they tried to show that $inf_y in F(||x-y||)= lambda $ exists.



by definition of $lambda$ it is the distance between $x$ and $F$










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
    $endgroup$
    – K.Power
    Mar 2 at 0:44











  • $begingroup$
    @K.Power actually I am looking for both
    $endgroup$
    – user515918
    Mar 2 at 0:46










  • $begingroup$
    If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
    $endgroup$
    – K.Power
    Mar 2 at 0:47










  • $begingroup$
    @K.Power I've been searching but I dont have any idea about the methode they using
    $endgroup$
    – user515918
    Mar 2 at 0:48










  • $begingroup$
    As these are very distinct questions you should definitely ask these in two separate posts.
    $endgroup$
    – K.Power
    Mar 2 at 0:50













4












4








4





$begingroup$


Let $E$ be $K-$vector space with a norm $|cdot|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set .



I am trying to show that any convergent sequence of elements $(x_n)_n in N$ of $F$ converge to an element $x in F$.



This means I need to show that $||x_n-x|| rightarrow 0$



I see that like a distance which is attended by the fuction $y rightarrow ||x-y||$ .



How can I show that $ x in F$?



I have been reading a solution which I don't really understand the intuition behind it :



let $x$ be in $E$ and let $B=$
after showing that its a compact, they tried to show that $inf_y in F(||x-y||)= lambda $ exists.



by definition of $lambda$ it is the distance between $x$ and $F$










share|cite|improve this question











$endgroup$




Let $E$ be $K-$vector space with a norm $|cdot|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set .



I am trying to show that any convergent sequence of elements $(x_n)_n in N$ of $F$ converge to an element $x in F$.



This means I need to show that $||x_n-x|| rightarrow 0$



I see that like a distance which is attended by the fuction $y rightarrow ||x-y||$ .



How can I show that $ x in F$?



I have been reading a solution which I don't really understand the intuition behind it :



let $x$ be in $E$ and let $B=$
after showing that its a compact, they tried to show that $inf_y in F(||x-y||)= lambda $ exists.



by definition of $lambda$ it is the distance between $x$ and $F$







functional-analysis normed-spaces topological-vector-spaces






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share|cite|improve this question













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edited Mar 2 at 21:55









K.Power

3,630926




3,630926










asked Mar 2 at 0:38









user515918user515918

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787







  • 1




    $begingroup$
    I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
    $endgroup$
    – K.Power
    Mar 2 at 0:44











  • $begingroup$
    @K.Power actually I am looking for both
    $endgroup$
    – user515918
    Mar 2 at 0:46










  • $begingroup$
    If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
    $endgroup$
    – K.Power
    Mar 2 at 0:47










  • $begingroup$
    @K.Power I've been searching but I dont have any idea about the methode they using
    $endgroup$
    – user515918
    Mar 2 at 0:48










  • $begingroup$
    As these are very distinct questions you should definitely ask these in two separate posts.
    $endgroup$
    – K.Power
    Mar 2 at 0:50












  • 1




    $begingroup$
    I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
    $endgroup$
    – K.Power
    Mar 2 at 0:44











  • $begingroup$
    @K.Power actually I am looking for both
    $endgroup$
    – user515918
    Mar 2 at 0:46










  • $begingroup$
    If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
    $endgroup$
    – K.Power
    Mar 2 at 0:47










  • $begingroup$
    @K.Power I've been searching but I dont have any idea about the methode they using
    $endgroup$
    – user515918
    Mar 2 at 0:48










  • $begingroup$
    As these are very distinct questions you should definitely ask these in two separate posts.
    $endgroup$
    – K.Power
    Mar 2 at 0:50







1




1




$begingroup$
I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
$endgroup$
– K.Power
Mar 2 at 0:44





$begingroup$
I am voting to close, because it is not at all clear to me what you are asking. Are you asking for the intuition behind closed sets, or are you asking how to show that a finite dimensional subspace is closed?
$endgroup$
– K.Power
Mar 2 at 0:44













$begingroup$
@K.Power actually I am looking for both
$endgroup$
– user515918
Mar 2 at 0:46




$begingroup$
@K.Power actually I am looking for both
$endgroup$
– user515918
Mar 2 at 0:46












$begingroup$
If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
$endgroup$
– K.Power
Mar 2 at 0:47




$begingroup$
If it is the latter then you will find such a question has been asked before on this site. I should also point out that you need to show that any convergent sequence in $F$ has its limit in $F$.
$endgroup$
– K.Power
Mar 2 at 0:47












$begingroup$
@K.Power I've been searching but I dont have any idea about the methode they using
$endgroup$
– user515918
Mar 2 at 0:48




$begingroup$
@K.Power I've been searching but I dont have any idea about the methode they using
$endgroup$
– user515918
Mar 2 at 0:48












$begingroup$
As these are very distinct questions you should definitely ask these in two separate posts.
$endgroup$
– K.Power
Mar 2 at 0:50




$begingroup$
As these are very distinct questions you should definitely ask these in two separate posts.
$endgroup$
– K.Power
Mar 2 at 0:50










6 Answers
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Let $xin E$ be the limit of a sequence $(x_n)_n$ where $x_nin F$ for all $n$. If $x=0$, nothing needs to be shown.
Once you know that $B:=,yin Fmid $ is compact, the continuous(!) map $BtoBbb R$, $ymapsto |y-x|$ attains its minimum at some $y_0in Bsubset F$. As $x_nto x$ and $|x|>0$, almost all $x_n$ are $in B$. Then from $|y_0-x|le |x_n-x|to 0$, we conclude $|y_0-x|=0$ and finally $x=y_0in F$.






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    $begingroup$

    Another approach will be to show that a finite-dimensional normed space (over $mathbb R$ or $mathbb C$ or indeed over any complete normed field) is complete. Then conclude that it is closed in a larger normed space.






    share|cite|improve this answer









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      Hint:



      1. Use that all norms on $F$ are equivalent to show that all bounded subsets of $F$ which are closed in $F$ are compact.


      2. Show that your sequence is contained in a bounded subset of $F$ which is closed in $F$.


      3. Find a subsequence which has a limit in $F$. What is the limit?






      share|cite|improve this answer











      $endgroup$




















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        To elaborate a bit on GEdgar's suggestion: We know that $F$ has a basis $e_1,dots,e_n$. Using Bolzano-Weierstrass you can show that we have the existence of some $r>0$ such that for any choice of scalars $c_1,dots,c_nin K$ we have
        $$|sum_i=1^nc_ie_i|geq rsum_i=1^n|c_i|.$$
        Using this you can show that $F$ is complete. Consider a cauchy sequence $(x_m)subset F$. We know that each $x_m=sum_i=1^nc_i^(m)e_i$. Using the inequality above and the cauchyness of $(x_m)$ try and show that each sequence of scalars $(c_i^(m))$ is also Cauchy. As $K$ is complete we know that these sequences each have a limit in $K$. Show that $(x_m)$ converges to the element of $F$ defined by these scalar limits.






        share|cite|improve this answer











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          I think K.Power's answer is the best elementary proof but if you don't know the inequality used in his/her answer, you can argue successfully without it:



          Let $F=textspane^1,cdots, e^n,$ where the $ langle e^i,e^jrangle =delta^ij; 1le ile n.$



          Now, suppose $Fsupset (x_n)to xin E$. Then, $x_i=a^i_1e^1+cdots +a^i_ne^n; 1le ile n.$



          Of course, $(x_n)$ is a Cauchy sequence in $E$, so since



          $|x_m-x_k|^2=|(a^m_1-a^k_1)e^1+cdots (a^m_n-a^k_n)e^n|^2=|a^m_1-a^k_1|^2+cdots +|a^m_n-a^k_n|^2$



          (because the $(e^i)$ are orthogonal), each $(a_i^m)$ is a Cauchy sequence in $K$.



          Assuming now that $K$ is complete, we have $a^m_ito a_iin K; 1le ile n$.



          To finish, consider the vector $x=a_1e^1+cdots +a_ne^n$.



          Clearly, $xin F$ and now, another application of the triangle inequality shows that



          $|x-x_n|to 0$, from which it follows that $x_nto xin F$






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          • $begingroup$
            Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
            $endgroup$
            – K.Power
            Mar 2 at 12:01










          • $begingroup$
            @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
            $endgroup$
            – Matematleta
            Mar 2 at 15:58










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            Ahh yes that's a neat argument
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            – K.Power
            Mar 2 at 19:56


















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          You haven't correctly stated what you need to prove -- you need to show that any sequence of elements of $F$ that converges in $E$ in fact has its limit in $F$.



          But I think it may be easier to show that the complement of F is open. Consider an element $x in E setminus F$. It has a distance $gamma gt 0$ from $F$, given by $||z||$, where $x = y+z$ with $y in F$ and $z perp F$. Then $B(x, gamma /2) subseteq E setminus F$ so $x$ is in the interior of $E setminus F$. We chose $x$ arbitrarily, so this proves $E setminus F$ is open so that $F$ is closed.






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          • $begingroup$
            how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
            $endgroup$
            – user515918
            Mar 2 at 1:14










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            Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
            $endgroup$
            – K.Power
            Mar 2 at 1:23










          • $begingroup$
            @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
            $endgroup$
            – Robert Shore
            Mar 2 at 16:21










          • $begingroup$
            Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
            $endgroup$
            – K.Power
            Mar 2 at 20:57










          • $begingroup$
            Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
            $endgroup$
            – Robert Shore
            Mar 3 at 3:05











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          6 Answers
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          6 Answers
          6






          active

          oldest

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          active

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          active

          oldest

          votes









          2












          $begingroup$

          Let $xin E$ be the limit of a sequence $(x_n)_n$ where $x_nin F$ for all $n$. If $x=0$, nothing needs to be shown.
          Once you know that $B:=,yin Fmid $ is compact, the continuous(!) map $BtoBbb R$, $ymapsto |y-x|$ attains its minimum at some $y_0in Bsubset F$. As $x_nto x$ and $|x|>0$, almost all $x_n$ are $in B$. Then from $|y_0-x|le |x_n-x|to 0$, we conclude $|y_0-x|=0$ and finally $x=y_0in F$.






          share|cite|improve this answer









          $endgroup$

















            2












            $begingroup$

            Let $xin E$ be the limit of a sequence $(x_n)_n$ where $x_nin F$ for all $n$. If $x=0$, nothing needs to be shown.
            Once you know that $B:=,yin Fmid $ is compact, the continuous(!) map $BtoBbb R$, $ymapsto |y-x|$ attains its minimum at some $y_0in Bsubset F$. As $x_nto x$ and $|x|>0$, almost all $x_n$ are $in B$. Then from $|y_0-x|le |x_n-x|to 0$, we conclude $|y_0-x|=0$ and finally $x=y_0in F$.






            share|cite|improve this answer









            $endgroup$















              2












              2








              2





              $begingroup$

              Let $xin E$ be the limit of a sequence $(x_n)_n$ where $x_nin F$ for all $n$. If $x=0$, nothing needs to be shown.
              Once you know that $B:=,yin Fmid $ is compact, the continuous(!) map $BtoBbb R$, $ymapsto |y-x|$ attains its minimum at some $y_0in Bsubset F$. As $x_nto x$ and $|x|>0$, almost all $x_n$ are $in B$. Then from $|y_0-x|le |x_n-x|to 0$, we conclude $|y_0-x|=0$ and finally $x=y_0in F$.






              share|cite|improve this answer









              $endgroup$



              Let $xin E$ be the limit of a sequence $(x_n)_n$ where $x_nin F$ for all $n$. If $x=0$, nothing needs to be shown.
              Once you know that $B:=,yin Fmid $ is compact, the continuous(!) map $BtoBbb R$, $ymapsto |y-x|$ attains its minimum at some $y_0in Bsubset F$. As $x_nto x$ and $|x|>0$, almost all $x_n$ are $in B$. Then from $|y_0-x|le |x_n-x|to 0$, we conclude $|y_0-x|=0$ and finally $x=y_0in F$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Mar 2 at 7:53









              Hagen von EitzenHagen von Eitzen

              283k23272507




              283k23272507





















                  2












                  $begingroup$

                  Another approach will be to show that a finite-dimensional normed space (over $mathbb R$ or $mathbb C$ or indeed over any complete normed field) is complete. Then conclude that it is closed in a larger normed space.






                  share|cite|improve this answer









                  $endgroup$

















                    2












                    $begingroup$

                    Another approach will be to show that a finite-dimensional normed space (over $mathbb R$ or $mathbb C$ or indeed over any complete normed field) is complete. Then conclude that it is closed in a larger normed space.






                    share|cite|improve this answer









                    $endgroup$















                      2












                      2








                      2





                      $begingroup$

                      Another approach will be to show that a finite-dimensional normed space (over $mathbb R$ or $mathbb C$ or indeed over any complete normed field) is complete. Then conclude that it is closed in a larger normed space.






                      share|cite|improve this answer









                      $endgroup$



                      Another approach will be to show that a finite-dimensional normed space (over $mathbb R$ or $mathbb C$ or indeed over any complete normed field) is complete. Then conclude that it is closed in a larger normed space.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Mar 2 at 1:20









                      GEdgarGEdgar

                      63.3k268172




                      63.3k268172





















                          2












                          $begingroup$

                          Hint:



                          1. Use that all norms on $F$ are equivalent to show that all bounded subsets of $F$ which are closed in $F$ are compact.


                          2. Show that your sequence is contained in a bounded subset of $F$ which is closed in $F$.


                          3. Find a subsequence which has a limit in $F$. What is the limit?






                          share|cite|improve this answer











                          $endgroup$

















                            2












                            $begingroup$

                            Hint:



                            1. Use that all norms on $F$ are equivalent to show that all bounded subsets of $F$ which are closed in $F$ are compact.


                            2. Show that your sequence is contained in a bounded subset of $F$ which is closed in $F$.


                            3. Find a subsequence which has a limit in $F$. What is the limit?






                            share|cite|improve this answer











                            $endgroup$















                              2












                              2








                              2





                              $begingroup$

                              Hint:



                              1. Use that all norms on $F$ are equivalent to show that all bounded subsets of $F$ which are closed in $F$ are compact.


                              2. Show that your sequence is contained in a bounded subset of $F$ which is closed in $F$.


                              3. Find a subsequence which has a limit in $F$. What is the limit?






                              share|cite|improve this answer











                              $endgroup$



                              Hint:



                              1. Use that all norms on $F$ are equivalent to show that all bounded subsets of $F$ which are closed in $F$ are compact.


                              2. Show that your sequence is contained in a bounded subset of $F$ which is closed in $F$.


                              3. Find a subsequence which has a limit in $F$. What is the limit?







                              share|cite|improve this answer














                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Mar 2 at 1:40

























                              answered Mar 2 at 1:26









                              triitrii

                              80817




                              80817





















                                  2












                                  $begingroup$

                                  To elaborate a bit on GEdgar's suggestion: We know that $F$ has a basis $e_1,dots,e_n$. Using Bolzano-Weierstrass you can show that we have the existence of some $r>0$ such that for any choice of scalars $c_1,dots,c_nin K$ we have
                                  $$|sum_i=1^nc_ie_i|geq rsum_i=1^n|c_i|.$$
                                  Using this you can show that $F$ is complete. Consider a cauchy sequence $(x_m)subset F$. We know that each $x_m=sum_i=1^nc_i^(m)e_i$. Using the inequality above and the cauchyness of $(x_m)$ try and show that each sequence of scalars $(c_i^(m))$ is also Cauchy. As $K$ is complete we know that these sequences each have a limit in $K$. Show that $(x_m)$ converges to the element of $F$ defined by these scalar limits.






                                  share|cite|improve this answer











                                  $endgroup$

















                                    2












                                    $begingroup$

                                    To elaborate a bit on GEdgar's suggestion: We know that $F$ has a basis $e_1,dots,e_n$. Using Bolzano-Weierstrass you can show that we have the existence of some $r>0$ such that for any choice of scalars $c_1,dots,c_nin K$ we have
                                    $$|sum_i=1^nc_ie_i|geq rsum_i=1^n|c_i|.$$
                                    Using this you can show that $F$ is complete. Consider a cauchy sequence $(x_m)subset F$. We know that each $x_m=sum_i=1^nc_i^(m)e_i$. Using the inequality above and the cauchyness of $(x_m)$ try and show that each sequence of scalars $(c_i^(m))$ is also Cauchy. As $K$ is complete we know that these sequences each have a limit in $K$. Show that $(x_m)$ converges to the element of $F$ defined by these scalar limits.






                                    share|cite|improve this answer











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                                      2





                                      $begingroup$

                                      To elaborate a bit on GEdgar's suggestion: We know that $F$ has a basis $e_1,dots,e_n$. Using Bolzano-Weierstrass you can show that we have the existence of some $r>0$ such that for any choice of scalars $c_1,dots,c_nin K$ we have
                                      $$|sum_i=1^nc_ie_i|geq rsum_i=1^n|c_i|.$$
                                      Using this you can show that $F$ is complete. Consider a cauchy sequence $(x_m)subset F$. We know that each $x_m=sum_i=1^nc_i^(m)e_i$. Using the inequality above and the cauchyness of $(x_m)$ try and show that each sequence of scalars $(c_i^(m))$ is also Cauchy. As $K$ is complete we know that these sequences each have a limit in $K$. Show that $(x_m)$ converges to the element of $F$ defined by these scalar limits.






                                      share|cite|improve this answer











                                      $endgroup$



                                      To elaborate a bit on GEdgar's suggestion: We know that $F$ has a basis $e_1,dots,e_n$. Using Bolzano-Weierstrass you can show that we have the existence of some $r>0$ such that for any choice of scalars $c_1,dots,c_nin K$ we have
                                      $$|sum_i=1^nc_ie_i|geq rsum_i=1^n|c_i|.$$
                                      Using this you can show that $F$ is complete. Consider a cauchy sequence $(x_m)subset F$. We know that each $x_m=sum_i=1^nc_i^(m)e_i$. Using the inequality above and the cauchyness of $(x_m)$ try and show that each sequence of scalars $(c_i^(m))$ is also Cauchy. As $K$ is complete we know that these sequences each have a limit in $K$. Show that $(x_m)$ converges to the element of $F$ defined by these scalar limits.







                                      share|cite|improve this answer














                                      share|cite|improve this answer



                                      share|cite|improve this answer








                                      edited Mar 2 at 1:49

























                                      answered Mar 2 at 1:44









                                      K.PowerK.Power

                                      3,630926




                                      3,630926





















                                          2












                                          $begingroup$

                                          I think K.Power's answer is the best elementary proof but if you don't know the inequality used in his/her answer, you can argue successfully without it:



                                          Let $F=textspane^1,cdots, e^n,$ where the $ langle e^i,e^jrangle =delta^ij; 1le ile n.$



                                          Now, suppose $Fsupset (x_n)to xin E$. Then, $x_i=a^i_1e^1+cdots +a^i_ne^n; 1le ile n.$



                                          Of course, $(x_n)$ is a Cauchy sequence in $E$, so since



                                          $|x_m-x_k|^2=|(a^m_1-a^k_1)e^1+cdots (a^m_n-a^k_n)e^n|^2=|a^m_1-a^k_1|^2+cdots +|a^m_n-a^k_n|^2$



                                          (because the $(e^i)$ are orthogonal), each $(a_i^m)$ is a Cauchy sequence in $K$.



                                          Assuming now that $K$ is complete, we have $a^m_ito a_iin K; 1le ile n$.



                                          To finish, consider the vector $x=a_1e^1+cdots +a_ne^n$.



                                          Clearly, $xin F$ and now, another application of the triangle inequality shows that



                                          $|x-x_n|to 0$, from which it follows that $x_nto xin F$






                                          share|cite|improve this answer











                                          $endgroup$












                                          • $begingroup$
                                            Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 12:01










                                          • $begingroup$
                                            @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                            $endgroup$
                                            – Matematleta
                                            Mar 2 at 15:58










                                          • $begingroup$
                                            Ahh yes that's a neat argument
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 19:56















                                          2












                                          $begingroup$

                                          I think K.Power's answer is the best elementary proof but if you don't know the inequality used in his/her answer, you can argue successfully without it:



                                          Let $F=textspane^1,cdots, e^n,$ where the $ langle e^i,e^jrangle =delta^ij; 1le ile n.$



                                          Now, suppose $Fsupset (x_n)to xin E$. Then, $x_i=a^i_1e^1+cdots +a^i_ne^n; 1le ile n.$



                                          Of course, $(x_n)$ is a Cauchy sequence in $E$, so since



                                          $|x_m-x_k|^2=|(a^m_1-a^k_1)e^1+cdots (a^m_n-a^k_n)e^n|^2=|a^m_1-a^k_1|^2+cdots +|a^m_n-a^k_n|^2$



                                          (because the $(e^i)$ are orthogonal), each $(a_i^m)$ is a Cauchy sequence in $K$.



                                          Assuming now that $K$ is complete, we have $a^m_ito a_iin K; 1le ile n$.



                                          To finish, consider the vector $x=a_1e^1+cdots +a_ne^n$.



                                          Clearly, $xin F$ and now, another application of the triangle inequality shows that



                                          $|x-x_n|to 0$, from which it follows that $x_nto xin F$






                                          share|cite|improve this answer











                                          $endgroup$












                                          • $begingroup$
                                            Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 12:01










                                          • $begingroup$
                                            @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                            $endgroup$
                                            – Matematleta
                                            Mar 2 at 15:58










                                          • $begingroup$
                                            Ahh yes that's a neat argument
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 19:56













                                          2












                                          2








                                          2





                                          $begingroup$

                                          I think K.Power's answer is the best elementary proof but if you don't know the inequality used in his/her answer, you can argue successfully without it:



                                          Let $F=textspane^1,cdots, e^n,$ where the $ langle e^i,e^jrangle =delta^ij; 1le ile n.$



                                          Now, suppose $Fsupset (x_n)to xin E$. Then, $x_i=a^i_1e^1+cdots +a^i_ne^n; 1le ile n.$



                                          Of course, $(x_n)$ is a Cauchy sequence in $E$, so since



                                          $|x_m-x_k|^2=|(a^m_1-a^k_1)e^1+cdots (a^m_n-a^k_n)e^n|^2=|a^m_1-a^k_1|^2+cdots +|a^m_n-a^k_n|^2$



                                          (because the $(e^i)$ are orthogonal), each $(a_i^m)$ is a Cauchy sequence in $K$.



                                          Assuming now that $K$ is complete, we have $a^m_ito a_iin K; 1le ile n$.



                                          To finish, consider the vector $x=a_1e^1+cdots +a_ne^n$.



                                          Clearly, $xin F$ and now, another application of the triangle inequality shows that



                                          $|x-x_n|to 0$, from which it follows that $x_nto xin F$






                                          share|cite|improve this answer











                                          $endgroup$



                                          I think K.Power's answer is the best elementary proof but if you don't know the inequality used in his/her answer, you can argue successfully without it:



                                          Let $F=textspane^1,cdots, e^n,$ where the $ langle e^i,e^jrangle =delta^ij; 1le ile n.$



                                          Now, suppose $Fsupset (x_n)to xin E$. Then, $x_i=a^i_1e^1+cdots +a^i_ne^n; 1le ile n.$



                                          Of course, $(x_n)$ is a Cauchy sequence in $E$, so since



                                          $|x_m-x_k|^2=|(a^m_1-a^k_1)e^1+cdots (a^m_n-a^k_n)e^n|^2=|a^m_1-a^k_1|^2+cdots +|a^m_n-a^k_n|^2$



                                          (because the $(e^i)$ are orthogonal), each $(a_i^m)$ is a Cauchy sequence in $K$.



                                          Assuming now that $K$ is complete, we have $a^m_ito a_iin K; 1le ile n$.



                                          To finish, consider the vector $x=a_1e^1+cdots +a_ne^n$.



                                          Clearly, $xin F$ and now, another application of the triangle inequality shows that



                                          $|x-x_n|to 0$, from which it follows that $x_nto xin F$







                                          share|cite|improve this answer














                                          share|cite|improve this answer



                                          share|cite|improve this answer








                                          edited Mar 2 at 17:30

























                                          answered Mar 2 at 2:14









                                          MatematletaMatematleta

                                          11.8k21020




                                          11.8k21020











                                          • $begingroup$
                                            Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 12:01










                                          • $begingroup$
                                            @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                            $endgroup$
                                            – Matematleta
                                            Mar 2 at 15:58










                                          • $begingroup$
                                            Ahh yes that's a neat argument
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 19:56
















                                          • $begingroup$
                                            Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 12:01










                                          • $begingroup$
                                            @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                            $endgroup$
                                            – Matematleta
                                            Mar 2 at 15:58










                                          • $begingroup$
                                            Ahh yes that's a neat argument
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 19:56















                                          $begingroup$
                                          Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 12:01




                                          $begingroup$
                                          Are you sure that you can just use the triangle inequality to show that each $(a_i^m)$ is Cauchy? You need to show that $|a_i^m-a_i^k|leq C|x_m-x_k|$ to use that $(x_m)$ is Cauchy. I think you need something like the inequality I used to show that.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 12:01












                                          $begingroup$
                                          @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                          $endgroup$
                                          – Matematleta
                                          Mar 2 at 15:58




                                          $begingroup$
                                          @K.Power: I assumed wlog orthogonality of the $e^i$ so I think the claim is ok.Or I guess you could use the max norm to avoid the problem entirely.
                                          $endgroup$
                                          – Matematleta
                                          Mar 2 at 15:58












                                          $begingroup$
                                          Ahh yes that's a neat argument
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 19:56




                                          $begingroup$
                                          Ahh yes that's a neat argument
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 19:56











                                          0












                                          $begingroup$

                                          You haven't correctly stated what you need to prove -- you need to show that any sequence of elements of $F$ that converges in $E$ in fact has its limit in $F$.



                                          But I think it may be easier to show that the complement of F is open. Consider an element $x in E setminus F$. It has a distance $gamma gt 0$ from $F$, given by $||z||$, where $x = y+z$ with $y in F$ and $z perp F$. Then $B(x, gamma /2) subseteq E setminus F$ so $x$ is in the interior of $E setminus F$. We chose $x$ arbitrarily, so this proves $E setminus F$ is open so that $F$ is closed.






                                          share|cite|improve this answer









                                          $endgroup$












                                          • $begingroup$
                                            how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                            $endgroup$
                                            – user515918
                                            Mar 2 at 1:14










                                          • $begingroup$
                                            Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 1:23










                                          • $begingroup$
                                            @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 2 at 16:21










                                          • $begingroup$
                                            Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 20:57










                                          • $begingroup$
                                            Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 3 at 3:05















                                          0












                                          $begingroup$

                                          You haven't correctly stated what you need to prove -- you need to show that any sequence of elements of $F$ that converges in $E$ in fact has its limit in $F$.



                                          But I think it may be easier to show that the complement of F is open. Consider an element $x in E setminus F$. It has a distance $gamma gt 0$ from $F$, given by $||z||$, where $x = y+z$ with $y in F$ and $z perp F$. Then $B(x, gamma /2) subseteq E setminus F$ so $x$ is in the interior of $E setminus F$. We chose $x$ arbitrarily, so this proves $E setminus F$ is open so that $F$ is closed.






                                          share|cite|improve this answer









                                          $endgroup$












                                          • $begingroup$
                                            how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                            $endgroup$
                                            – user515918
                                            Mar 2 at 1:14










                                          • $begingroup$
                                            Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 1:23










                                          • $begingroup$
                                            @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 2 at 16:21










                                          • $begingroup$
                                            Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 20:57










                                          • $begingroup$
                                            Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 3 at 3:05













                                          0












                                          0








                                          0





                                          $begingroup$

                                          You haven't correctly stated what you need to prove -- you need to show that any sequence of elements of $F$ that converges in $E$ in fact has its limit in $F$.



                                          But I think it may be easier to show that the complement of F is open. Consider an element $x in E setminus F$. It has a distance $gamma gt 0$ from $F$, given by $||z||$, where $x = y+z$ with $y in F$ and $z perp F$. Then $B(x, gamma /2) subseteq E setminus F$ so $x$ is in the interior of $E setminus F$. We chose $x$ arbitrarily, so this proves $E setminus F$ is open so that $F$ is closed.






                                          share|cite|improve this answer









                                          $endgroup$



                                          You haven't correctly stated what you need to prove -- you need to show that any sequence of elements of $F$ that converges in $E$ in fact has its limit in $F$.



                                          But I think it may be easier to show that the complement of F is open. Consider an element $x in E setminus F$. It has a distance $gamma gt 0$ from $F$, given by $||z||$, where $x = y+z$ with $y in F$ and $z perp F$. Then $B(x, gamma /2) subseteq E setminus F$ so $x$ is in the interior of $E setminus F$. We chose $x$ arbitrarily, so this proves $E setminus F$ is open so that $F$ is closed.







                                          share|cite|improve this answer












                                          share|cite|improve this answer



                                          share|cite|improve this answer










                                          answered Mar 2 at 1:11









                                          Robert ShoreRobert Shore

                                          3,603324




                                          3,603324











                                          • $begingroup$
                                            how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                            $endgroup$
                                            – user515918
                                            Mar 2 at 1:14










                                          • $begingroup$
                                            Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 1:23










                                          • $begingroup$
                                            @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 2 at 16:21










                                          • $begingroup$
                                            Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 20:57










                                          • $begingroup$
                                            Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 3 at 3:05
















                                          • $begingroup$
                                            how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                            $endgroup$
                                            – user515918
                                            Mar 2 at 1:14










                                          • $begingroup$
                                            Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 1:23










                                          • $begingroup$
                                            @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 2 at 16:21










                                          • $begingroup$
                                            Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                            $endgroup$
                                            – K.Power
                                            Mar 2 at 20:57










                                          • $begingroup$
                                            Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                            $endgroup$
                                            – Robert Shore
                                            Mar 3 at 3:05















                                          $begingroup$
                                          how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                          $endgroup$
                                          – user515918
                                          Mar 2 at 1:14




                                          $begingroup$
                                          how can we write x=y+z ? E is not an euclidien space to talk about the orthogonal of a subspace !
                                          $endgroup$
                                          – user515918
                                          Mar 2 at 1:14












                                          $begingroup$
                                          Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 1:23




                                          $begingroup$
                                          Aren't you assuming that the OP knows that finite dimensional subspaces are complemented? This implies they already know that they are closed.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 1:23












                                          $begingroup$
                                          @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                          $endgroup$
                                          – Robert Shore
                                          Mar 2 at 16:21




                                          $begingroup$
                                          @K.Power $F + x$ is finite dimensional so you can work there to come up with $z$.
                                          $endgroup$
                                          – Robert Shore
                                          Mar 2 at 16:21












                                          $begingroup$
                                          Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 20:57




                                          $begingroup$
                                          Do you mind elaborating? I still don't see how you can talk about $zperp F$ and knowing that $d(F,x)>0$ without knowing that $F$ is complemented.
                                          $endgroup$
                                          – K.Power
                                          Mar 2 at 20:57












                                          $begingroup$
                                          Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                          $endgroup$
                                          – Robert Shore
                                          Mar 3 at 3:05




                                          $begingroup$
                                          Not sure what you mean by “complemented.” I don’t recall the term from when I last studied linear algebra roughly three decades ago. But since $F$ is finite-dimensional, fix a basis for $F$, and for any $x notin F$ and use the basis created by $F+x$ to create an inner product, and therefore a norm, that’s equivalent to the existing norm.
                                          $endgroup$
                                          – Robert Shore
                                          Mar 3 at 3:05

















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