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We know that a linear operator [tex]T[/tex]:X[tex]\rightarrow[/tex]Y between two Banach Spaces X and Y is an open mapping if [tex]T[/tex] is surjective. Here open mapping means that [tex]T[/tex] sends open subsets of X to open subsets of Y.

Prove that if [tex]T[/tex] is an open mapping between two Banach Spaces then it is not necessarily a closed mapping, i.e. there could exist a closed subset of X that maps to a subset of Y which is not closed.

In other words, give a counter example. Being new to functional analysis, this has made me scratch my head..

Prove that if [tex]T[/tex] is an open mapping between two Banach Spaces then it is not necessarily a closed mapping, i.e. there could exist a closed subset of X that maps to a subset of Y which is not closed.

In other words, give a counter example. Being new to functional analysis, this has made me scratch my head..

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