known as Courant-Fisher Minimax theorem after the contributions of E. Fischer in the finite dimensional case [@Fis] and R. Courant in the case of hermitian compact operators on an infinite dimensional Hilbert space [@Cou]. Concretely, given a Hilbert space $H$ and a symmetric compact operator $A$ acting on $H$ with eigenvalues $(\lambda_n(A))_{n\in \mathbb{N}}$ (arranged in decreasing order and counting multiplicity) then for every natural $n$ we have that$$\begin{aligned}
 \lambda_n(A) &=\max_{M} \min_{\varphi\in M, \|\varphi\|=1} <A\varphi, \varphi> \notag  \\  & = \min_{N} \max_{\varphi\in N, \|\varphi\|=1} <A\varphi, \varphi>, \notag\end{aligned}$$ where $M$ runs over the subspaces of $H$ of dimension $n$ and $N$ runs over the subspaces of $H$ of codimension $n-1$.

Among the multiple consequences of the minimax principle we highlight Cauchy’s Interlacing Theorem and Weyl’s inequalities. The first result gives interesting inequalities between the eigenvalues of a Hermitian matrix A and those of a principal submatrix. H. Weyl showed the continuity of the eigenvalues in [@Wey], initiating a study of inequalities for eigenvalues and singular values which was continued, among others, by G. Polya, A. Horn and K. Fan. In particular the Ky Fan maximum principle [@Fan49] provides an extremum property of the sum of the first $k$ eigenvalues of a symmetric 