(All that follows assumes we are talking about a self-adjoint operator [itex]A[/itex] on a Hilbert space [itex]\mathscr H[/itex].) The first volume of Reed-Simon defines(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\mathscr H_{\rm pp} = \left\{ \psi \in \mathscr H: \mu_\psi \text{ is pure point} \right\}.

[/tex]

The book seems to take for granted that [itex]\mathscr H_{\rm pp}[/itex] is aclosedsubspace of [itex]\mathscr H[/itex], but this is not at all obvious to me. Can someone please explain why this is the case? Thanks in advance!

I suppose I'd also like to know the following: Is there anything in [itex]\mathscr H_{\rm pp}[/itex] that isnotan eigenvector for [itex]A[/itex]?

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# The pure-point subspace of a Hilbert space is closed

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