Isomorphism between two spaces, one of them is a Banach Space.
3
$begingroup$
Let $E$ be a Banach space. And $X$ be normed vector space.
If we have an isomorphism between $E$ and $X$. can we prove then that $X$ is also a Banach space ?
(In other words, does isomorphism conserves the Banach structure ? )
general-topology functional-analysis
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
$endgroup$
|
show 1 more comment
3
$begingroup$
Let $E$ be a Banach space. And $X$ be normed vector space.
If we have an isomorphism between $E$ and $X$. can we prove then that $X$ is also a Banach space ?
(In other words, does isomorphism conserves the Banach structure ? )
general-topology functional-analysis
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
$endgroup$
1
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
2
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
1
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34
|
show 1 more comment
3
3
3
$begingroup$
Let $E$ be a Banach space. And $X$ be normed vector space.
If we have an isomorphism between $E$ and $X$. can we prove then that $X$ is also a Banach space ?
(In other words, does isomorphism conserves the Banach structure ? )
general-topology functional-analysis
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
$endgroup$
Let $E$ be a Banach space. And $X$ be normed vector space.
If we have an isomorphism between $E$ and $X$. can we prove then that $X$ is also a Banach space ?
(In other words, does isomorphism conserves the Banach structure ? )
general-topology functional-analysis
general-topology functional-analysis
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
edited Dec 31 '18 at 19:45
Kenny Wong
18.5k21438
18.5k21438
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
asked Dec 31 '18 at 19:43
Anas BOUALIIAnas BOUALII
1267
1267
1
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
2
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
1
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34
|
show 1 more comment
1
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
2
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
1
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34
1
1
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
2
2
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
1
1
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34
|
show 1 more comment
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1
$begingroup$
When you say "isomorphism" do you mean isomorphism of vector spaces or isomorphism of topological vector spaces?
$endgroup$
– Eric Wofsey
Dec 31 '18 at 19:55
$begingroup$
topological vector space
$endgroup$
– Anas BOUALII
Dec 31 '18 at 19:57
2
$begingroup$
Check if your function is uniformly continuous in both directions (this is the case for bounded linear maps), then it preserves cauchy sequences and convergence sequences in both direction.
$endgroup$
– Math_QED
Dec 31 '18 at 20:02
$begingroup$
See here: math.stackexchange.com/questions/1639544/…
$endgroup$
– Math1000
Dec 31 '18 at 20:11
1
$begingroup$
@KaviRamaMurthy: Not a duplicate, since an isomorphism of topological vector spaces is not only a continuous map, but actually a homemorphism. So the counterexamples given in that answer don't apply, and the answer to this one is yes instead of no.
$endgroup$
– Nate Eldredge
Jan 1 at 5:34