Complemented subspaces conastructed from finite pieces- part II
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
asked 4 hours ago
user129564
874
add a comment |
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
asked 4 hours ago
user129564
874
add a comment |
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
asked 4 hours ago
user129564
874
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
asked 4 hours ago
user129564
874
asked 4 hours ago
user129564
874
asked 4 hours ago
user129564
874
asked 4 hours ago
user129564
874
asked 4 hours ago
user129564
874
874
add a comment |
add a comment |
1 Answer
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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.
answered 3 hours ago
Mikhail Ostrovskii
3,177927
add a comment |
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1 Answer
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1 Answer
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oldest
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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.
answered 3 hours ago
Mikhail Ostrovskii
3,177927
add a comment |
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.
answered 3 hours ago
Mikhail Ostrovskii
3,177927
add a comment |
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.
answered 3 hours ago
Mikhail Ostrovskii
3,177927
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.
answered 3 hours ago
Mikhail Ostrovskii
3,177927
answered 3 hours ago
Mikhail Ostrovskii
3,177927
answered 3 hours ago
Mikhail Ostrovskii
3,177927
answered 3 hours ago
Mikhail Ostrovskii
3,177927
3,177927
add a comment |
add a comment |
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