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I sense the completness of the normed spaces has a role, so I searched the incomplete metric spaces such as $c_{00}$ to find examples. And space must be infinite dimensional otherwise since every finite dimensional normed space is Banach I may not find an example. But the problem here I cannot arrange the mapping such that the mapping sends closed sets onto closed sets, For example polynomial space is not complete but I cannot understand the closed sets in this space(other than closed proper subsets). How can I construct such an example?

Jale'de jaled
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1 Answers1

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You can look at $X=c_{00}$ and take $T:c_{00}\to c_{00}$ given by $$T[(x_n)_{n\in\Bbb N}]=(2^n x_n)_{n\in\Bbb N}.$$ This is obviously linear and unbounded. Further it is a bijection as well as open as the map is bounded from below, so it is a closed map.

s.harp
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  • the problem that I cannot see the closedness, how did you conclude the boundedness below implies closedness? – Jale'de jaled Jun 16 '20 at 03:03
  • Boundedness below implies openness: $|x|≤ |Tx|$ so the image of any open ball around a point $x$ contains an open ball (here of the same radius) around $Tx$, hence $T$ sends neighbourhoods to neighbourhoods and is open. Now an open bijective map sends closed sets to closed sets, simply because if $C$ is closed then $X-T(C)=T(X-C)$ is open and then $T(C)$ must be closed. – s.harp Jun 16 '20 at 07:16