# The Space of Continuous Periodic Functions Is a Set of First Category inAP(X)

Authors: Zhe-Ming Zheng; Hui-Sheng Ding; Gaston M. N'Guérékata
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1], Bezandry and Diagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink [6], Levitan and Zhikov [7], N’Guérékata [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 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 byℝ the set of real numbers, by X a Banach space, and by C(ℝ,X) the set of all continuous functions f:ℝ→X.Definition 1 (see [4]).
A functionf∈C(ℝ,X) is called almost periodic if, for every ε>0, there exists l(ε)>0 such that every interval of length l(ε) contains a number τ with the property that
(1)supt∈ℝ∥f(t+τ)-f(t)∥<ε.
We denote the collection of all such functions by AP(X).Recall thatAP(X) is a Banach space under the supremum norm.Definition 2.
A functionf∈C(ℝ,X) is called periodic if there exists l>0 such that
(2)f(t+l)=f(t),∀t∈ℝ.
Here, l is called a period of f. We denote the collection of all such functions by P(X). For f∈P(X), we call l0 the fundamental period if l0 is the smallest period of f.Remark 3.
Similar to the proof in [4, page 1], it is not difficult to show that if f∈P(X) is not constant, and then f has the fundamental period.Definition 4 (see [8]).
A functionf∈C(ℝ,X) is called almost automorphic if, for every real sequence (sm), there exists a subsequence (sn) such that
(3)g(t)=limn→∞f(t+sn)
is well defined for each t∈ℝ and
(4)limn→∞g(t-sn)=f(t)
for each t∈ℝ. Denote by AA(X) the set of all such functions.Recall that there exists an almost automorphic function which is not almost periodic, for instance, the following function:(5)f(t)=sin12+cost+cos2t,t∈ℝ.Before the proof of our main results, we need to recall the notion about the first category.Definition 5 (see [14]).
LetS be a topological space. A set E⊂S is said to be nowhere dense if its closure has an empty interior. The sets of the first category in S are those that are countable unions of nowhere dense sets. Any subset of S that is not of the first category is said to be of the second category in S.Theorem 6.
P
(
X
) is a set of first category in AP(X).Proof.
Forn=1,2,…, we denote
(6)Pn={f∈C(ℝ,X):thereexistsl∈[n,n+1]suchthatf(t+l)=f(t)∀t∈ℝ}.
Then, it is easy to see that(7)P(X)=⋃n=1∞‍Pn.
We divide the remaining proof into two steps.
Step  1. Every Pn is a closed subset of AP(X).
Letf∈AP(X)∖Pn. Then, for every l∈[n,n+1], there exists tl∈ℝ such that f(tl+l)≠f(tl). Denote
(8)εl:=14∥f(tl+l)-f(tl)∥>0,l∈[n,n+1].
In addition, due to the continuity of f, for every l∈[n,n+1], there exists δl>0 such that
(9)∥f(tl+s)-f(tl)∥≥3εl,∀s∈(l-δl,l+δl).
Obviously, we have
(10)[n,n+1]⊂⋃l∈[n,n+1]‍(l-δl,l+δl).
Then, by the Heine-Borel theorem, there exists l1,…,lk∈[n,n+1] such that
(11)[n,n+1]⊂⋃i=1k‍(li-δli,li+δli),
where k is a fixed positive integer. Letting ε=min1≤i≤k{εli}, and
(12)N(f,ε):={g∈AP(X):∥g-f∥AP(X)<ε},
for every g∈N(f,ε), we claim that g∉Pn. In fact, for every l∈[n,n+1], there exists i∈{1,…,k} such that
(13)l∈(li-δli,li+δli).
Then, by (9), we have
(14)∥f(tli+l)-f(tli)∥≥3εli≥3ε,
which yields that
(15)∥g(tli+l)-g(tli)∥≥∥f(tli+l)-f(tli)∥-∥f(tli+l)-g(tli+l)∥-∥f(tli)-g(tli)∥≥3ε-ε-ε=ε>0,
where ∥g-f∥AP(ℝ)<ε was used. So, we know that N(f,ε)⊂AP(X)∖Pn, which means that Pn is a closed subset of AP(X).
Step  2. Every Pn has an empty interior.
It suffices to prove that, for everyf∈Pn and δ>0, N(f,δ)⋂‍(AP(X)∖Pn)≠∅. Now let f∈Pn and δ>0. In the following, we discuss two cases.
Case I. f is constant.
We denote(16)fδ(t)=cost+cos(2t)3·δ·x0+f(t),t∈ℝ,
wherex0∈X is some constant with ∥x0∥=1. Then fδ∈N(f,δ) and fδ∉Pn since fδ is not periodic.
Case II. f is not constant.
By Remark3, f has a fundamental period l0. We denote
(17)fδ(t)=f(t)+f(tπ)·δMf,t∈ℝ,
where Mf=supt∈ℝ∥f(t)∥. Obviously, fδ∈N(f,δ). Also, we claim that fδ∉Pn. In fact, if this is not true, then there exists T∈[n,n+1] such that
(18)fδ(t+T)=fδ(t),t∈ℝ,
that is,
(19)f(t+T)+f(t+Tπ)·δMf=f(t)+f(tπ)·δMf,t∈ℝ.
Let
(20)F1(t)=f(t+T)-f(t),F2(t)=δMf[f(tπ)-f(t+Tπ)],t∈ℝ.
Then F1(t)≡F2(t). If F1(t)≡F2(t)≡C, where C is a fixed constant, then
(21)f(t+T)=f(t)+C,t∈ℝ,
which yields
(22)C=f(kT)-f(0)k→0,k→∞,
since f is bounded. Thus, we have
(23)f(t+T)=f(t),f(tπ)=f(t+Tπ),t∈ℝ.
Noting that l0 is the fundamental period of f and πl0 is the fundamental period of f(·/π), there exist two positive integers pandq such that
(24)pl0=T=qπl0,
that is, π=p/q, which is a contradiction. If F1=F2 is not constant, then, by Remark 3, we can assume that T0 is the fundamental period of F1 and F2. Noting that l0 is a period of F1 and πl0 is a period of F2, similar to the above proof, we can also show that π is a rational number, which is a contradiction.
In conclusion,P(X) is countable unions of closed subsets with empty interior. So P(X) is a set of first category.Remark 7.
SinceAP(X) is a set of second category, it follows from Theorem 6 that AP(X)∖P(X) 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.
A
P
(
X
) is a set of first category in AA(X).Proof.
Firstly,AP(X) is a closed subset of AA(X). Secondly, AP(X) has an empty interior in AA(X). In fact, letting
(25)φ(t)=sin12+cost+cos(2t),t∈ℝ,
for every f∈AP(X) and δ>0, we have fδ∉AP(X) and fδ∈N(f,δ), where
(26)fδ(t)=f(t)+δx0Mφφ(t),t∈ℝ,Mφ=supt∈ℝ∥φ(t)∥,
and x0∈X is some constant with ∥x0∥=1. This completes the proof.Remark 9.
By Theorem8, AA(X)∖AP(X) is a set of second category in AA(X), 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.



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