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Is there a simple example of a linear map $T: X \to Y$ between Banach spaces that is not closed, i.e. if $x_n \to 0$ in $X$ and $T(x_n) \to y$ in $Y$, then $y=0$? I know the Closed Graph Theorem says that if $T: X \to Y$ is a linear map between Banach spaces, then \begin{align*} T \text{ is bounded } \iff T \text{ is closed } \iff \text{gr}(T) \text { is a closed subspace of } X \oplus Y. \end{align*} So I would have to find an unbounded linear map between Banach spaces, correct? What exactly would that be?

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We can construct an unbounded linear functional easily. Pick your favorite basis $e_n$ with $\lVert e_n \rVert = 1$ of $X$ (which can be done with the axiom of choice) and define $f$ as $$f(x)=\sum_n n x_n,\,\,\,\,\text{where }\,\,\,\,x=\sum_n x_n e_n.$$ Note that this is well-defined, as $x_n$ is non-zero for finitely many $n$ for every $x$. Then pick $$y_n = \frac{e_n}{\sqrt{n}},$$ which clearly goes to $0$, but $f(y_n)=\sqrt{n} \to +\infty$.

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  • $\begingroup$ What exactly would $X$ and $Y$ be? Are there any concrete Banach spaces that this works for? $\endgroup$
    – MATH-LORD
    Mar 1, 2021 at 21:37
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    $\begingroup$ @MATH-LORD Pick your favorite Banach space $X$ over the field $\mathbb{F}$ and $Y=\mathbb{F}$. Note that you can't construct this function "exactly", but you can prove that it exists. $\endgroup$
    – Botond
    Mar 1, 2021 at 21:39
  • $\begingroup$ Alright, thank you for the help! $\endgroup$
    – MATH-LORD
    Mar 1, 2021 at 21:41

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