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Im reading Peter Lax book and he says: For any subset $S \subset X'$, we define $S^\perp$ as the subset of those vectors in $X$ that are annihilated by every vector in S. This confuses me a bit, shoudent it be every functional in X'' that vanishes on S? Or is this the same thing by identifying those vectors in X?

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    Which of his books? $S^\bot=\bigcap_{f\in S} Ker(f)$ makes sense to me.2012-12-25
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    You have two options: the annihilator in $X$ Lax defines or the one in $X''$ which you propose. If $X$ is not reflexive those are distinct in general. Consider $S = \{0\}$ for an easy example.2012-12-25
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    Oki good! thanks!2012-12-25
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    Some books, like *Functional Analysis* by Conway, use different notation for the annihilator $S^{\perp}$ and the pre-annihilator ${}^{\perp}\!S$ to avoid such confusion.2013-01-01

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