Some Classes of Function Spaces, Their Properties, and ApplicationsView this Special Issue
The Space of Continuous Periodic Functions Is a Set of First Category in
We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.
Since the last century, the study on almost periodic type functions and their applications to evolution equations has been of great interest for many mathematicians. There is a large literature on this topic. Several books are especially devoted to almost periodic type functions and their applications to differential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse , Bezandry and Diagana , Bohr , Corduneanu , Diagana , Fink , Levitan and Zhikov , N’Guérékata [8, 9], Pankov , Shen and Yi , Zaidman , and Zhang .
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 difficult 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
Throughout the rest of this paper, we denote by the set of real numbers, by a Banach space, and by the set of all continuous functions .
Definition 1 (see ). A function is called almost periodic if, for every , there exists such that every interval of length contains a number with the property that We denote the collection of all such functions by .
Recall that is a Banach space under the supremum norm.
Definition 2. A function is called periodic if there exists such that Here, is called a period of . We denote the collection of all such functions by . For , we call the fundamental period if is the smallest period of .
Remark 3. Similar to the proof in [4, page 1], it is not difficult to show that if is not constant, and then has the fundamental period.
Definition 4 (see ). A function is called almost automorphic if, for every real sequence , there exists a subsequence such that is well defined for each and for each . Denote by the set of all such functions.
Recall that there exists an almost automorphic function which is not almost periodic, for instance, the following function:
Before the proof of our main results, we need to recall the notion about the first category.
Definition 5 (see ). Let be a topological space. A set is said to be nowhere dense if its closure has an empty interior. The sets of the first category in are those that are countable unions of nowhere dense sets. Any subset of that is not of the first category is said to be of the second category in .
Theorem 6. is a set of first category in .
Proof. For , we denote
Then, it is easy to see that We divide the remaining proof into two steps.
Step??1. Every is a closed subset of .
Let . Then, for every , there exists such that . Denote In addition, due to the continuity of , for every , there exists such that Obviously, we have Then, by the Heine-Borel theorem, there exists such that where is a fixed positive integer. Letting , and for every , we claim that . In fact, for every , there exists such that Then, by (9), we have which yields that where was used. So, we know that , which means that is a closed subset of .
Step??2. Every has an empty interior.
It suffices to prove that, for every and , . Now let and . In the following, we discuss two cases.
Case I. is constant.
where is some constant with . Then and since is not periodic.
Case II. is not constant.
By Remark 3, has a fundamental period . We denote where . Obviously, . Also, we claim that . In fact, if this is not true, then there exists such that that is, Let Then . If , where is a fixed constant, then which yields since is bounded. Thus, we have Noting that is the fundamental period of and is the fundamental period of , there exist two positive integers such that that is, , which is a contradiction. If is not constant, then, by Remark 3, we can assume that is the fundamental period of and . Noting that is a period of and is a period of , similar to the above proof, we can also show that is a rational number, which is a contradiction.
In conclusion, is countable unions of closed subsets with empty interior. So is a set of first category.
Remark 7. Since is a set of second category, it follows from Theorem 6 that is a set of second category, which means that, to some extent, the “amount” of almost periodic functions (not periodic) is far more than the “amount” of continuous periodic functions.
Theorem 8. is a set of first category in .
Proof. Firstly, is a closed subset of . Secondly, has an empty interior in . In fact, letting for every and , we have and , where and is some constant with . This completes the proof.
Remark 9. By Theorem 8, is a set of second category in , which means that, to some extent, the “amount” of almost automorphic functions (not almost periodic) is far more than the “amount” of almost periodic functions.
H.-S. Ding acknowledges support from the NSF of China (11101192), the Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), and the Jiangxi Provincial Education Department (GJJ12173).
C. Corduneanu, Almost Periodic Functions, Chelsea Publishing, New York, NY, USA, 2nd edition, 1989.
W. X. Shen and Y. F. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, no. 647 of Memoirs of the American Mathematical Society, 1998.