Let be a symmetric, hyperbolic bilinear form (that is, of signature ) over a finite dimensional real vector space (though this may hold in general as well). If is positive, meaning , then for all we have

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# The reverse Cauchy-Schwarz inequality

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Let be a symmetric, hyperbolic bilinear form (that is, of signature ) over a finite dimensional real vector space (though this may hold in general as well). If is positive, meaning , then for all we have

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One is reduced to show that for vectors and scalars subject to , we have . One can rewrite the above as a quadratic polynomial in , which has discriminant precisely , whence the result.

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