\leq W \sqrt{\ell} \sqrt{\sum_{i=1}^{n}\norm{\bx_{i}}^{2}},
    \end{aligned}$$ where in the last inequality we just used the Jensen’s inequality for the function $t \mapsto \sqrt{t}$. The claim of the Lemma now follows from the definition of Rademacher Complexity.

Plugging in the result of Lemma \[eq:l2\_bound\] into Lemma \[lem: help\_lem01\] and consequently plugging the implication of the latter into the general result of Theorem \[thm:main\_theorem01\] we deduce the following general result.

\[eq:rad\_comp\_ell\_2\] Let the function class $\mathcal{F}$ be as given in Equation with $B=\{\norm{f}_{2} \leq W\}$. Assume $\norm{X}_{2} \leq K$ and $\abs{Y} \leq R$, $\mathbb{P}_{X}$ and $\mathbb{P}_{Y}$-almost surely. Then the following upper bounds for the empirical and true Rademacher complexities of the class $\mathcal{F}$ hold: $$\begin{aligned}
    \hat{\mathfrak{R}}_{\mathbb{S}}\paren{\mathcal{F}} & \leq  
    W \sqrt{\frac{\ell}{n}} \paren{\sqrt{2\tr{\hat{\Sigma}}} + 4\sqrt{\norm[1]{\hat{\Sigma}}_{op}}\sqrt{\log(enD)}};
                            \\
    {\mathfrak{R}}_{n}\paren{\mathcal{F}} &\leq W \sqrt{\frac{\ell}{n}} \paren{\sqrt{2\tr{\Sigma}} + 4  \sqrt{\ee[1]{}{\norm[1]{\hat{\Sigma}}_{op}}} \sqrt{\log(enD)} }.
$$ where we denote the operator norm of a matrix as $\norm{\cdot}_{op}$. Furthermore, with probability at least $1-\delta$ it holds: $$\begin{aligned}
            L\paren{f} - L_{n}\paren{f} \leq 4\paren{R+F} \paren{ W\sqrt{\frac{\ell}{n}}\paren{\sqrt{2\tr \hat{\Sigma}} + 4 \sqrt{\norm[2]{\hat{\Sigma}}_{op} }\sqrt{\log\paren{enD}} } + \paren{R+2F}\sqrt{\frac{\log\paren{\frac{2}{\delta}}}{2n}}},
    \end{aligned}$$ or similarly in data-independent form (when using bound on Rademacher complexity instead of its data-dependent proxy) we have: $$\begin{aligned}
                L\paren{f} - L_{n}\paren{f} \leq 4\paren{R+F} \paren{  W \sqrt{\frac{\ell}{n}} \paren{\sqrt{2\tr{\Sigma}} + 4  \sqrt{\ee[1]{}{\norm[1]{\hat{\Sigma}}_{op}}} \sqrt{\log(enD)} 