with $rank(g)=n$. By Corollary \[c gram schmidt\] there exists a (minimal) tripotent $v\in \mathcal{U}_2(g)$ which is orthogonal to $f$, thus $v$ belongs to $\mathcal{U}_0(f)$. Having in mind that $v\in \mathcal{U}_2(g)$ is also orthogonal to every $v_i$ with $i>n$ we have that $$\|x\|_{v}=\|\sum_{i=1}^n \lambda_i(x) v_i \|_{v} \geq \|\sum_{i=1}^n \lambda_n(x) v_i \|_{v} = \lambda_n(x)\|g \|_{v} = \lambda_n(x),$$ which finishes the proof.

The following result is a generalization of the classical Weyl’s inequality obtained in [@Wey]. More concretely, we shall show that the distance between the corresponding singular values of two elements in a weakly compact JB$^*$-triple is bounded by the distance between the elements.

\[t Weyl’s Inequality distance between singular values\] Let $\mathcal{U}$ be a weakly compact JB$^*$-triple. Let $x, y$ be two elements in $\mathcal{U}$ with singular values $(\lambda_i(x))_{i\in \mathbb{N}}$ and $(\lambda_i(y))_{i\in \mathbb{N}}$, respectively. Then $$\label{eq Weyl inequality}  \sup_{n\in \mathbb{N}} |\lambda_n(x) - \lambda_n(y)|\leq \|x-y\|.$$

By Theorem \[t minmax\] we have that whenever $e$ runs over the set of tripotents in $\mathcal{U}$ of rank $n$, $$\lambda_n(x) =\max_e \inf\{\|x\|_{v}: v\in \mathcal{U}_2(e) \hbox{ minimal tripotent}  \}=$$ $$\max_e \inf \{\|y-(y-x)\|_{v}: v\in \mathcal{U}_2(e) \hbox{ minimal tripotent} \}\leq$$ $$\max_e \inf \{\|y\|_{v}: v\in \mathcal{U}_2(e) \hbox{ minimal tripotent} \} + \|y-x\|=\lambda_n(y)+\|y-x\|$$ and our statement follows straightforwardly.

Given a JB$^*$-triple, 