Example of a linear onto map which is not open

Consider the identity $$I : (\ell^1, \|\cdot\|_1) \to (\ell^1, \|\cdot\|_\infty)$$ which is a continuous bijection since $\|\cdot\|_\infty \le \|\cdot\|_1$. However, it is not open.

Indeed, if $I$ were open, it would imply that $I^{-1}$ is bounded i.e. that $\|\cdot\|_1$ is bounded by $\|\cdot\|_\infty$, which is false.

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Functional Analysis

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