Let $X$ be a Banach space, $\overline{}$ be its closed unit ball and $$ its unit sphere. A retraction from $\overline{}$ to $$ is a continuous map $r:\overline{}→$ such that $\left.r\right|_$ is the identity map on $$. When $X$ has finite dimension, no such retraction exists. This statement is wrong in any infinite dimensional space $X$.
The following exercise gives an example.
Let $X$ be $C^0[0,1]$ equipped with supremum norm.
(a) Let $A : X → X$ be defined by $$ A f(t)={|f(t)+1-2 t(1-{‖f‖})|}-1+2 t(1-{‖f‖}) . $$ Show that $A$ is the identity on $$ and ${‖Af‖}≥\frac17$ for $f∈\overline{}$.
(b) Deduce that there is a retraction from $\overline{}$ to $$.
My attempt:
(a) Argue by contradiction, suppose $\|Af\|<\frac17$ then $\frac17>Af(t)$ for all $t\in[0,1]$.
Taking $t=1$,
$$\frac17>Af(1)\ge-1+2 (1-{\|f\|})$$So ${‖f‖}>\frac37$.
I don't know how to proceed, can you help me?
The hint is to consider $t_1<t_2$ for which $f\left(t_1\right)=-\frac17$ and $f\left(t_2\right)=-\frac37$.
I don't see why $t_1$ exist.