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 ? )










share|cite|improve this question











$endgroup$








  • 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
















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 ? )










share|cite|improve this question











$endgroup$








  • 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














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 ? )










share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 31 '18 at 19:45









Kenny Wong

18.5k21438




18.5k21438










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














  • 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










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