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Let $X,Y$ be Banach spaces. A map $T: X \to Y$ is said to be closed if: $$ x_n \to x \ \land \ Tx_n \to y \implies Tx = y$$

Do you have an example of a linear map that does not satisfy this condition? And is there are a reason that we (well, my book) define it as a map between Banach spaces, not just normed spaces?

JustANoob
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    The definition makes perfect sense in normed spaces. The reason for introducing it in the Banach space setting is probably because they want to get to the Closed-Graph theorem, which is an important result in functional analysis, where the completeness is important (so for the sake of presentation, everything is restricted to the Banach space setting). – peek-a-boo Jan 07 '22 at 10:07
  • If I am not mistaken, a linear map between Banach spaces is closed if and only if it is continuous. So the notion of closed linear map is more interesting if defined on non-complete spaces. – Dirk Jan 07 '22 at 10:36

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Just take any non-continuous linear map on a Banach space, an example of a non-continuous functional (so image $\Bbb R$) can be found here.

On a non-Banach space (for $X$ a Banach space we get that continuity and closedness are equivalent, by the Closed Graph Theorem, hence the previous example) you can consider $X=C_p([0,1])$ (the continuous real functions on $[0,1]$ in the pointwise topology, a locally convex topological vector space but not normable) and $T(f)=\int_0^1 f(x)dx \in \Bbb R$ as a "natural" example.

Martin Sleziak
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Henno Brandsma
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