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Eingegangen am 9. Miirz 1965) DIAz, J. B. and F. T. METCALF Math. Annalen 162, 120-139 {1965) Complementary Inequalities ID: Inequalities Complementary to Schwarz's Inequality in Hilbert Space* Dedicated to Professor GoTTFRIED KoTHE on the occasion of his sixtieth birthday, December 25, 1965 By J. B. Duz and F. T. METCALF** in College Park, Maryland Introduction The Schwarz inequality [18, p. 198] in a Hilbert space (with real or complex scalars) implies that I(Ax, Bx)i2 ~ (Ax, Ax) (Bx, Bx), where xis an arbitrary vector of the Hilbert space, A and Bare operators on the Hilbert space to itself.

Dann ist

Es werde nun gezeigt: Die Zusammenziehung wird durch die heiden Verformungen T,A, die jedes A in T 1 A iiberfiihrt, und (1 - s) T 1 A + slf' A 0 , bei der T 1 A in If' A 0 iibergeht, geleistet. Es bleibt dabei offenbar nur noch zu beweisen, daB (1- s) T 1 A + slf' A 0 immer in