\subsection{Proof of Lemma \ref{lem:norm inequality}}
We first note that orthogonal projections are bounded linear operators. Therefore, $\mathcal{P}_{\mathcal{S}_\mathcal{X}}$'s are bounded and linear. One can easily show that compositions of bounded linear operators are also bounded and linear. Therefore, $L\mathcal{P}_{\mathcal{S}_\mathcal{X}}$ is bounded and linear for any orthogonal projection $\mathcal{P}_{\mathcal{S}_\mathcal{X}}$. Then, for $p\geq 1$, we have
\begin{eqnarray*}
    \|\mathcal{P}^\dagger_{\mathcal{S}_\mathcal{X}}L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_p} &\overset{a}{\leq}& \|\mathcal{P}_{\mathcal{S}_\mathcal{X}}^\dagger\|_{S_\infty} \|L^\dagger L\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_p} 
    \\ &\overset{b}{\leq}& \|\mathcal{P}_{\mathcal{S}_\mathcal{X}}^\dagger\|_{S_\infty} \|L^\dagger L\|_{S_p} \|\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_\infty}
    \\ &=&  \|L^\dagger L\|_{S_p}, 
\end{eqnarray*}
where $(a)$ and $(b)$ follow from Hölder's inequality. Note that $\|\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_\infty}$ is the standard operator norm on $\mathcal{H}$, i.e., $\|\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_\infty}= \max_{\|x\|\leq 1}\|\mathcal{P}_{\mathcal{S}_\mathcal{X}} x\|$. Since $\mathcal{P}_{\mathcal{S}_\mathcal{X}}$ is an orthogonal projection, we have $\|\mathcal{P}_{\mathcal{S}_\mathcal{X}}\|_{S_\infty}=\|\mathcal{P}_{\mathcal{S}_\mathcal{X}}^\dagger\|_{S_\infty}=1$.
