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I'm reading "A short course on Operator Semigroups" by Engel and Nagel. I came across a result about core of an operator and there is a part in which I'm stuck.

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On page 41, when Hahn Banach Theorem is applied, why is inner product used? We are just working on an arbitrary Banach Space. There hasn't been any mention of Hilbert spaces, so far.

Any help would be appreciated.

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    The notation is meant to mimic an inner product, but it is not one. For $x\in X$, $x'\in X'$, we write $\langle x,x'\rangle=x'(x)$. – 2017-02-05
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    Aah. Makes sense now. Thanks! – 2017-02-05

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