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Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach space?

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    $\begingroup$ Is this a homework exercise? $\endgroup$ Nov 13, 2022 at 17:37
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    $\begingroup$ I do not know. It just came in my mind. However, I could not solve it. $\endgroup$
    – Anupam
    Nov 13, 2022 at 18:06

1 Answer 1

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No. The closed graph theorem in this form is equivalent to $X$ being a barreled space. See item 15 here.

There are incomplete normed spaces that are barreled. See here.

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