Hilbert–Schmidt operators are compact

For every $x\in H$, [F2] gives $\mathrm{\parallel }{P}_{F}x{\mathrm{\parallel }}^2={\sum }_{e\in F}\mathrm{\mid }⟨x,e⟩{\mathrm{\mid }}^2\le \mathrm{\parallel }x{\mathrm{\parallel }}^2$ and $P_{F}x\in \mathrm{span}\{e:e\in F\}$; thus $P_F$ is linear by construction, bounded with $\mathrm{\parallel }{P}_F\mathrm{\parallel }\le 1$, and its range lies in $\mathrm{span}\{e:e\in F\}$.

Since $F$ is finite, [F5] fixes $n\in \mathbb{N}$ and a bijection $\sigma$ from $n$ onto $F$; the list $e_{\sigma (0)},\dots ,e_{\sigma (n-1)}$ is injective because the family is orthonormal, and its image spans $Z:=\mathrm{span}\{e:e\in F\}$, so it is an ordered basis of $Z$ of finite length; therefore ${\bar{B}}_Z$ is compact by [F5].

The set $P_F({\bar{B}}_H)$ is contained in ${\bar{B}}_Z$ by [step 1.1], since $\mathrm{\parallel }{P}_{F}x\mathrm{\parallel }\le \mathrm{\parallel }x\mathrm{\parallel }\le 1$ and $P_{F}x\in Z$; the restriction of $T$ to $Z$ is continuous by [F7], so $T({\bar{B}}_Z)$ is compact by [step 1.2] and [F6]; as $T{P}_F({\bar{B}}_H)=T(P_F({\bar{B}}_H))\subseteq T({\bar{B}}_Z)$, its closure is a closed subset of the compact set $T({\bar{B}}_Z)$, hence compact by [F6], and $T{P}_F$ is compact by [F1].

For finite $F\subseteq G\subseteq E$ and $x\in H$ we have $P_F{P}_{G}x=P_{F}x$, because $⟨P_{G}x,e⟩=⟨x,e⟩$ for $e\in G$; hence $(I-P_F)P_{G}x=P_{G}x-P_{F}x={\sum }_{e\in G\setminus F}⟨x,e⟩e$ and, by linearity of $T$, $T(I-P_F)P_{G}x={\sum }_{e\in G\setminus F}⟨x,e⟩Te$; the triangle inequality and the finite Cauchy–Schwarz inequality give $\mathrm{\parallel }T(I-P_F)P_{G}x\mathrm{\parallel }\le ({\sum }_{e\in G\setminus F}\mathrm{\mid }⟨x,e⟩{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}({\sum }_{e\in G\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}\le \mathrm{\parallel }x\mathrm{\parallel }({\sum }_{e\in E\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$, where the last step uses [F2] for the coefficient factor and [F4] for the tail factor.

As $G$ runs over the finite subsets of $E$ containing $F$, the net $(I-P_F)P_{G}x=(P_{G}x-P_{F}x)$ converges to $(I-P_F)x$, by [F3] and the boundedness of $P_F$ from [step 1.1]; the continuous operator $T$ of [F7] therefore carries this net to a net converging to $T(I-P_F)x=(T-T{P}_F)x$, while [step 2.2] bounds every term of that net by $\mathrm{\parallel }x\mathrm{\parallel }({\sum }_{e\in E\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$; the norm being continuous, the limit obeys the same bound, and taking the supremum over $\mathrm{\parallel }x\mathrm{\parallel }\le 1$ gives $\mathrm{\parallel }T-T{P}_F\mathrm{\parallel }\le ({\sum }_{e\in E\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$.

Given a real $\epsilon >0$, [F4] provides a finite $F\subseteq E$ with ${\sum }_{e\in E\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2<{\epsilon }^2$; then $\mathrm{\parallel }T-T{P}_F\mathrm{\parallel }<\epsilon$ by [step 3.1], and $T{P}_F$ is compact by [step 2.1], so for every positive tolerance there is a compact operator $T{P}_F$ within that tolerance of $T$.

By [F9] applied to the countably many nonempty sets of finite $F\subseteq E$ satisfying ${\sum }_{e\in E\setminus F}\mathrm{\parallel }Te{\mathrm{\parallel }}^2<(n+1{)}^{-2}$ for $n\in \mathbb{N}$ — each nonempty by [F4] — there is a sequence $(F_n)$ of finite subsets of $E$ with these tails; then $\mathrm{\parallel }T-T{P}_{F_n}\mathrm{\parallel }\le (n+1{)}^{-1}\to 0$ by [step 3.1], and each $T{P}_{F_n}$ is compact by [step 2.1].

The target $K$ is a Banach space by [F8], so the norm limit $T$ of the compact operators $T{P}_{F_n}$ is compact by [F8]; this proves claim 3, while claims 1 and 2 are [step 1.1] with [step 2.1] and [step 3.1] with [step 4.1].