Min-max theorem
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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
- <math>\lambda_{-1}\le\lambda_{-2}\le\cdots<0<\lambda_1<\lambda_2<\cdots,<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.