Min-max theorem

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

Let A be a compact, Hermitian operator on a Hilbert space H. Recall that the eigenvalues of such an operator form a sequence of real numbers whose only possible cluster point is zero. Let the nonzero eigenvalues of A be

<math>\lambda_{-1}\le\lambda_{-2}\le\cdots<0<\cdots\le\lambda_2\le\lambda_1</math>

with multiplicity taken into account. Then

<math>\lambda_n=\inf_{H_{n-1}\subset H}\sup_{x\perp H_{n-1}}\frac{(Ax,x)}{\|x\|^2},</math>

<math>\lambda_{-n}=\sup_{H_{n-1}\subset H}\inf_{x\perp H_{n-1}}\frac{(Ax,x)}{\|x\|^2},</math>

where <math>H_{n-1}</math> is taken over all (n−1)-dimensional subspaces of H.