Rend. Istit. Mat. Univ. Trieste Vol. XL, 29–44 (2008) Exponents in R of Elements in a Uniformly Complete Φ-Algebra Mohamed Ali Toumi Abstract. In this paper we give a new and constructive proof of exponents in R of elements in a uniformly complete Φ-algebra. As an application we establish the Young’s inequal- ity and we give a short proof of the H¨ older’s inequality on uniformly complete Φ-algebras. Keywords: p-Power, H¨older’s Inequality, Cauchy-Schwartz’s Inequality, f-Algebras, Uniformly Complete Φ-Algebra, Young’s Inequality MS Classification 2000: 06F25; 26D15; 47B65 1. Introduction The proofs as well as extensions and applications of the well-known H¨older’s inequality can be found in many works about real functions, analysis, functional analysis, L P -spaces ([5–7, 11]). This inequality involved exponents in the field R of real numbers. To the best of our knowledge, there is not works dealing with the subject of exponents in the field R of elements in uniformly complete Φ-algebras, that is uniformly complete f -algebras with an identity element, except the paper of J. L. Krivine [10], which relies heavily on the representation theory and the Axiom of Choice. In this paper we discuss and give a new and purely algebraic proof of exponents in R of elements in uniformly complete Φ-algebras. As an application we establish the Young’s inequality and we give a short proof of the H¨older’s inequal- ity on uniformly complete Φ-algebras. We take it for granted that the reader is familiar with the notions of vector lattices (or Riesz spaces) and operators between them. For terminology, notations and concepts that are not explained in this paper we refer to the standard monographs [1, 9] and [15].