International Mathematical Forum, 5, 2010, no. 21, 1001 - 1013 The Riesz Theorem and α-n-Norms in Random n-Normed Spaces B.Surender Reddy Department of Mathematics, PGCS, Saifabad Osmania University, Hyderabad-500004, A.P., India bsrmathou@yahoo.com Abstract The primary purpose of this paper is to prove the Riesz theorem in random n-normed space as a generalization of linear n-normed space. Also we study some properties of random n-norm and introduce a con- cept of random anti n-norm. Mathematics Subject Classification: 46A19, 46B20, 46B99, 46C05 Keywords: Riesz theorem, random n-norm, random α-n-norm, random n-compact set 1 Introduction Gahler [2] introduced the theory of n-norm on a linear space. Gunawan and Mashadi [5], Kim and Cho [9], Malceski [10] and Misiak [11] developed the theory of n-normed space. In [12] K. Menger introduced the notion of probabilistic metric spaces. The idea of K. Menger was to use distribution function instead of non negative real numbers as values of the metric. The concept of random normed spaces were introduced by Serstnev [16] and [8]. Golet [4] introduced the concept of random 2-norm on a linear space. The concept of random n-normed spaces was introduced by Iqbal H. Jebril [6]. Riesz [15] obtained the Riesz theorem in a normed space. Park and Chu [13] have extended the Riesz theorem in a normed space to n-normed linear space. In this paper, we extend the Riesz theorem in n-normed linear space to random n-normed space. Also we study some properties of random n-norm and introduce a concept of random anti n-norm.