Complemented subspaces conastructed from finite pieces- part II












2














This is a follow up to: Complemented subspace constructed from finite pieces



Suppose $Y=overline{cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_nsubseteq E_{n+1}$. Can one conclude that $Y$ is complemented in $X$?



In light of the answer to the previous question, a related question would be the following:



Is $c_0$ complemented in every separable subspace of $l_infty$ that contains it. I suspect the answer is no, but cannot think of a counterexample.






fa.functional-analysis banach-spaces







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    2














    This is a follow up to: Complemented subspace constructed from finite pieces



    Suppose $Y=overline{cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_nsubseteq E_{n+1}$. Can one conclude that $Y$ is complemented in $X$?



    In light of the answer to the previous question, a related question would be the following:



    Is $c_0$ complemented in every separable subspace of $l_infty$ that contains it. I suspect the answer is no, but cannot think of a counterexample.






    fa.functional-analysis banach-spaces







    share|cite|improve this question

























      2












      2








      2







      This is a follow up to: Complemented subspace constructed from finite pieces



      Suppose $Y=overline{cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_nsubseteq E_{n+1}$. Can one conclude that $Y$ is complemented in $X$?



      In light of the answer to the previous question, a related question would be the following:



      Is $c_0$ complemented in every separable subspace of $l_infty$ that contains it. I suspect the answer is no, but cannot think of a counterexample.






      fa.functional-analysis banach-spaces







      share|cite|improve this question













      This is a follow up to: Complemented subspace constructed from finite pieces



      Suppose $Y=overline{cup E_n}$ is a closed subspace of a separable Banach space X, where each $E_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E_nsubseteq E_{n+1}$. Can one conclude that $Y$ is complemented in $X$?



      In light of the answer to the previous question, a related question would be the following:



      Is $c_0$ complemented in every separable subspace of $l_infty$ that contains it. I suspect the answer is no, but cannot think of a counterexample.






      fa.functional-analysis banach-spaces




      fa.functional-analysis banach-spaces






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      user129564

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          The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.



          The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.



          The answer is positive if the space is reflexive - you can consider the weak limit of the projections.






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            The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.



            The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.



            The answer is positive if the space is reflexive - you can consider the weak limit of the projections.






            share|cite|improve this answer


























              3














              The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.



              The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.



              The answer is positive if the space is reflexive - you can consider the weak limit of the projections.






              share|cite|improve this answer
























                3












                3








                3






                The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.



                The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.



                The answer is positive if the space is reflexive - you can consider the weak limit of the projections.






                share|cite|improve this answer












                The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.



                The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces, Ark. Mat. 18 (1980), no. 1, 101–106.



                The answer is positive if the space is reflexive - you can consider the weak limit of the projections.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Mikhail Ostrovskii

                3,177927




                3,177927






























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