Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 275702, 3 pages http://dx.doi.org/10.1155/2013/275702 Research Article The Space of Continuous Periodic Functions Is a Set of First Category in () Zhe-Ming Zheng, 1 Hui-Sheng Ding, 1 and Gaston M. N’Guérékata 2 1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2 Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA Correspondence should be addressed to Hui-Sheng Ding; dinghs@mail.ustc.edu.cn Received 17 January 2013; Accepted 12 February 2013 Academic Editor: J´ ozef Bana´ s Copyright © 2013 Zhe-Ming Zheng et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove that the space of continuous periodic functions is a set of frst category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of frst category in the space of almost automorphic functions. 1. Introduction Since the last century, the study on almost periodic type func- tions and their applications to evolution equations has been of great interest for many mathematicians. Tere is a large literature on this topic. Several books are especially devoted to almost periodic type functions and their applications to diferential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse [1], Bezandry and Diagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink [6], Levitan and Zhikov [7], N’Gu´ er´ ekata [8, 9], Pankov [10], Shen and Yi [11], Zaidman [12], and Zhang [13]. Although almost periodic functions have a very wide range of applications now, it seems that giving an example of almost periodic (not periodic) functions is more difcult than giving an example of periodic functions. Also, there is a similar problem for almost automorphic functions. In this paper, we aim to compare the “amount” of almost periodic functions (not periodic) with the “amount” of continuous periodic functions, and we also discuss the related problems for almost automorphic functions. 2. Main Results Troughout the rest of this paper, we denote by R the set of real numbers, by  a Banach space, and by (R,) the set of all continuous functions : R →. Defnition 1 (see [4]). A function ∈(R,) is called almost periodic if, for every >0, there exists () > 0 such that every interval of length () contains a number  with the property that sup ∈R     (+)−()     < . (1) We denote the collection of all such functions by (). Recall that () is a Banach space under the supremum norm. Defnition 2. A function  ∈ (R,) is called periodic if there exists >0 such that (+)=(), ∀∈ R. (2) Here,  is called a period of . We denote the collection of all such functions by (). For  ∈ (), we call  0 the fundamental period if  0 is the smallest period of . Remark 3. Similar to the proof in [4, page 1], it is not difcult to show that if  ∈ () is not constant, and then  has the fundamental period.