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If A and B are closed linear operators from $X$ to $X$ ($X$ is a normed vector space and the domain of them is X), is $A+B$ a closed operator? I think it's not but I can't find a counterexample. In other books, it is said to have a closed graph.

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    $\begingroup$ See math.stackexchange.com/q/53954 for a counterexample $\endgroup$ – user70600 Apr 2 '13 at 14:04
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    $\begingroup$ If $X$ is a Banach space, and if the domain of $A$ is $X$, then $A$ closed implies $A$ continuous (=bounded) by the closed graph theorem. The converse is obvious. So $A:X\longrightarrow X$ is closed if and only if it is continuous. In this case, $A+B$ will automatically be continuous, hence closed. So you need $X$ to be non complete in the first place. Then see the link provided by jim. $\endgroup$ – Julien Apr 2 '13 at 14:13

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