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How do I find that example of a discontinuous linear operator A from a Banach space to a normed vector space such that A has a closed graph?

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    The energy operator $[Hf](x)=-\frac{1}{2}\frac{d^2}{dx^2} f(x) + \frac{1}{2}x^2f(x)$ is an example. (But, as explained [here](http://en.wikipedia.org/wiki/Hellinger-Toeplitz_theorem), it is not defined on the whole domain $L^2(\mathbb R)$. Otherwise, it would contradict the CGT.)2012-03-06
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    The Closed Graph Theorem is about mappings between Banach spaces. The OP asks about a map from a Banach space into a normed space.2012-03-06
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    Let B be some infinite dimensional Banach space with Hamel basis $\{e_\alpha:\alpha\in\mathcal{A}\}$. Fix $\alpha_0\in\mathcal{A}$ and consider linear functional $$f:B\to\mathbb{C}:x\mapsto x_{\alpha_0},$$ where $x_{\alpha_0}$ is a $\alpha_0$-coordinate of $x$ in this basis.2012-03-06

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