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If $T$ is a continuous linear transformation of Banach space $B$ onto Banach space $B'$, then

  1. $T$ is an open mapping.
  2. $T$ is a closed mapping.
  3. $T$ is open as well as closed mapping.
  4. None of these.

It is an objective Question (with single correct answer) asked in an exam. I directly found from Open Mapping Theorem that its 1st option is correct. But, I found many sources on internet that are saying that $T$ is closed also. Please clear my doubt.

bipin
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    $T$ need not be closed. – Evangelopoulos Foivos Dec 08 '21 at 19:05
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    Are you maybe mixing up maps which are closed in the sense of topology (i.e. they map closed sets to closed sets) with closed operators in the sense of functional analysis (i.e. linear operators having a closed graph)? Regarding the latter definition, a linear operator between $F$-spaces has a closed graph if and only if that operator is bounded (continuous) - this is the so called "Closed Graph Theorem". – Thomas Dec 08 '21 at 19:39
  • I have not mixed anything. I just wrote question as given the the examination. – bipin Dec 09 '21 at 02:14
  • @Evangelopoulos F. any example? – bipin Dec 09 '21 at 02:16
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    See this https://math.stackexchange.com/questions/163844/is-an-open-linear-map-closed-to-some-extent – Evangelopoulos Foivos Dec 09 '21 at 06:50
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    'I have not mixed up anything' - I was referring to your statement 'I found many sources on internet that are saying that $T$ is closed also'. Or was this part of the question from the examination, also? If the 'sources on the internet' were referring to mappings with closed graphs, then the answer is 'yes'. Maybe these sources were mixing up closed maps with maps with closed graph. The example of @EvangelopoulosF. shows that 2. is false if you are referring to closed maps in the first sense. – Thomas Dec 09 '21 at 12:16

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