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The follow question arose from the paper: The Hundal Example Revisited

We consider a separable Hilbert Space $X$ with countable ortho-normal basis $\{e_n\}_{n=1}^\infty$. The following is an excerpt from page 4 of [1]:

... the hyperplane $\ker(e_1)$ and...

My question is regarding the notation $\ker(e_1)$. Since it is described as a hyperplane I can only assume that it must be defined, for $a\in X$, as followed: $\ker(a) := \{x\in X: \langle a,x \rangle = 0 \} .$ I am familiar with using $\ker(T)$ as the kernel of an operator $T$, but have not seen it used for elements of the space. I guess we could view $\operatorname{ker}(a)$ as an abbreviation for $\ker(\langle a,\cdot\rangle)$, viewing $\langle a,\cdot\rangle$ as a mapping from $X$ to $\mathbb F$ ($X$'s scalar field).

Is my definition and motivation correct? If not, could someone provide a correct version?

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    Thanks you both for your helpful comments.2012-01-10

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