I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like to know if there is a geometric proof, which does not need a background in Gaussian processes.

Let $H$ be a separable Hilbert space with ONB $(e_n)_{n \in \mathbb{N}}$ and let $(\alpha_n)_{n \in \mathbb{N}}$ be a bounded sequence with $\alpha_n>0$. We define the Ellipse $$ E_\alpha := \left\{x \in H \ \middle| \ \sum\limits_{n=1}^\infty \frac{|\langle x,e_n \rangle_H|^2}{\alpha_n^2} \leq 1 \right\} \ . $$ Section 2.9.3 together with Theorem 2.10.2 would imply the equivalence of \begin{align*} &a) \ \sum\limits_{n=1}^\infty \alpha_n^2 < \infty\\ &b) \ \exists (g_m)_{m \in \mathbb{N}} \subset H : \ E_\alpha \subset \overline{sco}(g_m \mid m \in \mathbb{N}) \ \text{ and } \ \log(m)||g_m||_H^2 \xrightarrow{m \rightarrow \infty} 0 \ . \end{align*} This seems like an interesting statement concerning geometry in Hilbert spaces. Can someone point me to some literature where questions of when ellipses are in the convex hulls of sequences? Or better yet, does someone know of a way to prove the equivalence with geometric methods? Any help is much appreciated.

I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like to know if there is a geometric proof, which does not need a background in Gaussian processes.

Let $H$ be a separable Hilbert space with ONB $(e_n)_{n \in \mathbb{N}}$ and let $(\alpha_n)_{n \in \mathbb{N}}$ be a bounded sequence with $\alpha_n>0$. We define the Ellipse $$ E_\alpha := \left\{x \in H \ \middle| \ \sum_{n=1}^\infty \frac{|\langle x,e_n \rangle_H|^2}{\alpha_n^2} \leq 1 \right\} \ . $$ Section 2.9.3 together with Theorem 2.10.2 would imply the equivalence of \begin{align*} &a) \ \sum\limits_{n=1}^\infty \alpha_n^2 < \infty\\ &b) \ \exists (g_m)_{m \in \mathbb{N}} \subset H : \ E_\alpha \subset \overline{\operatorname{sco}}(g_m \mid m \in \mathbb{N}) \ \text{ and } \ \log(m)\|g_m\|_H^2 \xrightarrow{m \rightarrow \infty} 0 \ . \end{align*} This seems like an interesting statement concerning geometry in Hilbert spaces. Can someone point me to some literature where questions of when ellipses are in the convex hulls of sequences? Or better yet, does someone know of a way to prove the equivalence with geometric methods? Any help is much appreciated.