such that $ x=\sum \lambda_i(x) u_i$ ([@BunChu92 Remark 4.6]). The real numbers $(\lambda_i(x))$ are called the *singular values* of $x$, and they are precisely the singular values of the function $x$ when considered as an element of (the commutative C$^*$-algebra) $\mathcal{U}_x$, i.e. the set of all the singular values coincide with the triple spectrum of $x$. We will refer to the sum $ \sum \lambda_i(x) u_i$ as an *atomic decomposition* of $x$. When the singular values vanishes at some natural (and thereafter) we can choose to remove all these summands or just consider the corresponding tripotents to be zero. The reader should be aware that while the sequence of singular values is uniquely determined, there are many different choices of the minimal tripotents the moment we have a singular value with multiplicity greater than one.

\[l innequality in minimax theorem\] Let $\mathcal{U}$ be a weakly compact JB$^*$-triple. Let $x$ be an element in $\mathcal{U}$ with an atomic decomposition $x= \sum_{i\geq 1} \lambda_i (x) v_i$. Then, for every natural $n$ and for every $e\in \mathcal{U}$ tripotent of rank $n$ there exists a minimal tripotent $v$ in $\mathcal{U}_2(e)$ such that $$\|x\|_v\leq \lambda_n(x).$$

For $n=1$ we have that $\|x\|_v\leq\|x\|=\lambda_1(x)$, for every minimal tripotent 