Anal. Theory Appl. Vol. 25, No. 2 (2009), 117–124 DOI10.1007/s10496-009-0117-9 SIMPLE PROOFS OF THE CAUCHY-SCHWARTZ INEQUALITY AND THE NEGATIVE DISCRIMINANT PROPERTY IN ARCHIMEDEAN ALMOST f -ALGEBRAS Adel Toumi, Mohamed Ali Toumi and Nedra Toumi (Facult ´ e des Sciences de Bizerte, Tunisia ) Received Apr. 10, 2008 Abstract. The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in archimedean almost f -algebras [5] , based on a sequence approximation. Key words: Cauchy-Schwartz inequality, f-algebra, almost f-algebra, positive square ℓ- algebra AMS (2000) subject classification: 06F25, 47B65 1 Introduction The standard negative discriminant property states as follows. Let A be a real ℓ-algebra and let a, b, c ∈ A. If λ 2 a + 2λ b + c ≥ 0 for all λ ∈ R, then b 2 ≤ ac. The standard application of the negative discriminant property is, of course, the classical Cauchy-Schwartz inequality. This idea was immediately used by C. B. Huijsmans and B. de Pagter, see Proposition 3.3 and Corollary 3.5 in [6], to prove that a positive linear map T from an archimedean f -algebra A into a semiprime f -algebra B satisfies the Cauchy-Schwartz inequality, that is: (T (ab)) 2 ≤ T ( a 2 ) T ( b 2 ) (1) for all a, b ∈ A. Moreover, S. J. Bernau and C. B. Huijsmans [2] were able to verify the inequality (1) away with the semiprimeness condition. They did show, however, that any archimedean f -algebra satisfies the negative discriminant property, and by the way they deduced that the inequality (1) remains true on archimedean f -algebras. Although, their proof is constructive, the way to do that is too long and quite involved. Recently, G. Buskes and A. Van Rooij [5] , took