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I am searching a reference to following counter-example for closed mapping theorem in normed spaces:

Given $X$ a Banach space, and $Y$ a normed space that is not a Banach space, Let $T: X \to Y$ to be a linear operator such that $graf(T)$ is a closed set and $T$ is an unbounded operator.

Martin Sleziak
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Joao Maia
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1 Answers1

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Take any infinite dimensional Banach space $(X,\|\cdot\|)$, a discontinuous linear functional (this needs the axiom of choice) $f:X\to \mathbb R$, and define $\|x\|_1=\|x\|+|f(x)|$. Then $Y=X$ endowed with $\|\cdot\|_1$ is a normed space, and the identical map $T:X\to Y$ is unbounded (since $f$ is unbounded) with closed graph (because the inverse is continuous).

Jochen
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  • for future people looking at this question: a discussion of the existance of discontinuous functionals can be found here -- https://math.stackexchange.com/questions/99206/discontinuous-linear-functional – yoshi Apr 18 '18 at 02:20