# Show that $T:\,c_0\to c_0\;\;$, $x\mapsto T(x)=(1,x_1,x_2,\cdots),$ has no fixed points

As a follow-up to my previous question Show that $$T:\,c_0\to c_0\;\;$$, $$x\mapsto T(x)=(1,x_1,x_2,\cdots),$$ is non-expansive. Let $$X$$ be a normed linear space and $$X=c_0$$ (the space of sequences of real numbers which converge to $$0$$). Define \begin{align} T:\,&c_0\to c_0 \\ &x\mapsto T(x)=(1,x_1,x_2,\cdots), \end{align} for arbitrary $$x=(1,x_1,x_2,\cdots)\in c_0.$$ I want to show that $$T$$ has no fixed points.

Remark: We have that

$$c_0=\{\bar{x}=(x_1,x_2,\cdots) :x_n\to 0\;\text{as}\;n\to \infty\}.$$

MY TRIAL

Suppose for contradiction that $$T$$ has fixed points in $$X$$, then there exists $$u\in c_0$$ s.t. $$T(u)=u.$$ That is, \begin{align} T(u_1,u_2,\cdots)=(u_1,u_2,\cdots), \end{align} Please, how do I draw out a contradiction from this?

• which book of this question can you suggest me..please – Inverse Problem Mar 24 at 4:38
• @Inverse Problem: I got this from my the book "C.E. Chidume, Applicable Functional Analysis, Ibadan University Press Publishing House, 2014, University of Ibadan, Ibadan, Nigeria. ISBN: 978-978-8456-31-5". There are several other books too! – Omojola Micheal Mar 25 at 21:56
• ........can you tell me...math.stackexchange.com/questions/3159556/… math.stackexchange.com/questions/3160256/… which have this questions? do you have idea – Inverse Problem Mar 26 at 15:30
• Applicable Functional Analysis u have this book? – Inverse Problem Mar 26 at 15:37
• @Inverse Problem: Yes, I do! – Omojola Micheal Mar 27 at 18:12

$$(1,u_1,u_2,...)=(u_1,u_2,..)$$ implies $$1=u_1,u_1=u_2$$ etc so $$u_n=1$$ for all $$n$$. But then $$(u_n) \notin c_0$$.
• Sorry Sir, how's $u_n\notin c_0$? – Omojola Micheal Dec 14 '18 at 8:58
• Let me guess, it's because $u_n$ does not go to zero and $n\to\infty.$ Therefore, $u_n\notin c_o.$ I'm I right? – Omojola Micheal Dec 14 '18 at 9:01
• Yes, $u_n \to 1$. – Kavi Rama Murthy Dec 14 '18 at 9:02