Min-max theorem

In mathematics, the min-max theorem is an important result in the theory of hilbert spaces.

Let A be a compact, Hermitian operator on a Hilbert space H. Let the nonzero eigenvalues of A be

$\lambda _{-1}\leq \lambda _{-2}\leq \cdots <0<\lambda _{1}<\lambda _{2}<\cdots ,$

with multiplicity taken into account. Then

$\lambda _{n}=\inf _{H_{n-1}\subset H}\sup _{x\perp H_{n-1}}{\frac {(Ax,x)}{\|x\|^{2}}},$

$\lambda _{-n}=\sup _{H_{n-1}\subset H}\inf _{x\perp H_{n-1}}{\frac {(Ax,x)}{\|x\|^{2}}},$

where $H_{n-1}$ is taken over all (n-1)-dimensional subspaces of H.