$C_0$ direct sum of complex matrices
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How many elements in the $C_0$ direct sum of complex matrices,say$oplus_n M_n(mathbb{C})$? $oplus_n M_n(mathbb{C})={(x_n)in prod_n M_n(mathbb{C}):|x_n| to 0}$
I think there are uncountably many elements in $oplus_n M_n(mathbb{C})$,but one professor told me that it was not true.Why?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
mathrookie
649412
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show 3 more comments
up vote
-1
down vote
favorite
How many elements in the $C_0$ direct sum of complex matrices,say$oplus_n M_n(mathbb{C})$? $oplus_n M_n(mathbb{C})={(x_n)in prod_n M_n(mathbb{C}):|x_n| to 0}$
I think there are uncountably many elements in $oplus_n M_n(mathbb{C})$,but one professor told me that it was not true.Why?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
mathrookie
649412
What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago
|
show 3 more comments
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
How many elements in the $C_0$ direct sum of complex matrices,say$oplus_n M_n(mathbb{C})$? $oplus_n M_n(mathbb{C})={(x_n)in prod_n M_n(mathbb{C}):|x_n| to 0}$
I think there are uncountably many elements in $oplus_n M_n(mathbb{C})$,but one professor told me that it was not true.Why?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
mathrookie
649412
How many elements in the $C_0$ direct sum of complex matrices,say$oplus_n M_n(mathbb{C})$? $oplus_n M_n(mathbb{C})={(x_n)in prod_n M_n(mathbb{C}):|x_n| to 0}$
I think there are uncountably many elements in $oplus_n M_n(mathbb{C})$,but one professor told me that it was not true.Why?
operator-theory operator-algebras c-star-algebras von-neumann-algebras
operator-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
mathrookie
649412
asked yesterday
mathrookie
649412
asked yesterday
mathrookie
649412
asked yesterday
mathrookie
649412
asked yesterday
mathrookie
649412
649412
What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago
|
show 3 more comments
What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago
What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago
|
show 3 more comments
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What did you try so far ? What about $mathbb C$ itself ?
– André S.
yesterday
I think we can let the n-th elment be 1,others are 0,there are uncountably many elements.
– mathrookie
23 hours ago
I don't understand your approach. The collection of elements you describe is countable. Note that $mathbb C$ is uncountable already.
– André S.
22 hours ago
I do not understand the question. Any $C$-vector space is (at least) uncountable
– Adrián González-Pérez
21 hours ago
@André S.You mean $mathbb{C}$ in $oplus_{c_0}M_n(mathbb{C})$?
– mathrookie
19 hours ago