Footnote

... category1.1
A Category $ \mathscr{C}$ is said to be locally small iff its collection of morphisms form a proper set 
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... spectra)2.1
the condition of the spectrum being discrete implies that given a function $ f:\sigma(\hat{A})\rightarrow{\mathchoice {\setbox0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ then $ \sigma(f(\hat{A}))=f(\sigma(\hat{A}))$ 
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...\space 2.2
The reason behind the identification of objects of D with set of homomorphism from W to $ \{0,1\}$ is the following: by definition, a dual of D is a map $ D^*=D\rightarrow{\mathchoice {\setbox0=\hbox{$\displaystyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm R$}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$. We know from definition of spectral projectors that $ \hat{P}=\hat{P^2}$ for all $ P\in W$. This implies that the only values that $ \hat{P}$ can have are 1 or 0 therefore $ D^*=D\rightarrow\{0,1\}$ 
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... $ (A\in\Delta)$2.3
Proposition $ \lq\lq A\in\Delta''$ means that the value of A lies in some interval $ \Delta$ 
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... Theorem2.4
(Spectral Theorem) [3], [9], [4], states the following: suppose $ \hat{A}$ is a compact self-adjoint operator on a Hilbert space $ \mathcal{H}$ then, there is an orthonormal basis of $ \mathcal{H}$ consisting of eigenvectors of $ \hat{A}$. Each eigenvalue of $ \hat{A}$ is real. 
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... small3.1
A "small category" is a category whose collection of morphisms and objects form a genuine Set. 
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... algebra3.2
A boolean algebra S is an orthocomplemented and distributive lattice 
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... join3.3
Given two elements $ \hat{P}_{a_2}$ and $ \hat{P}_{a_3}$ of a lattice, the least upper bound (suprimum, join) is the unique element above $ \hat{P}_{a_2}$ and $ \hat{P}_{a_3}$ with respect to the partial ordering such that there is no other element in between. The least upper bound or join is denoted by $ \hat{P}_{a_2}\vee\hat{P}_{a_3}$ 
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