equivalent statements to statement that sphere is not contractible

Let $V$ be a normed space. Recall the definition of the sphere and the ball in $V$:

$$𝕊=\{v\in V;\parallel v\parallel =1\};𝔹=\{v\in V;\parallel v\parallel \le 1\}.$$

Proposition. The following are equivalent:

$(1)$ $𝕊$ is not contractible;

$(2)$ for each continous map $F:𝔹\to 𝔹$ there exists $x\in 𝔹$ such that $F(x)=x$;

$(3)$ there is no retraction from $𝔹$ onto $𝕊$.

Proof. The proof of this proposition probably can be found in some books about topology. I present here the proof from my lecture due to Prof. $\mathrm{G}\acute{\mathrm{o}}\mathrm{rniewicz}$.

$(1)\Rightarrow (2)$ Assume there exists a continous map $F:𝔹\to 𝔹$ such that for each $x\in 𝔹$ we have $F(x)\ne x$. Define a map $H:𝕊\times [0,1]\to 𝕊$ as follows:

Thanks to the condition $F(x)\ne x$ this map is well defined and it is easy to check that this is a homotopy from the identity map to constant map. But $𝕊$ is not contractible. Contradiction.

$(2)\Rightarrow (3)$ Assume there exists a retraction $r:𝔹\to 𝕊$. Define a map $F:𝔹\to 𝔹$ by the formula $F(x)=-r(x)$. This map has no fixed point. Contradiction.

$(3)\Rightarrow (1)$ Assume that $𝕊$ is contractible and take any homotopy $H:𝕊\times [0,1]\to 𝕊$ from constant map to identity map, i.e. for all $x\in 𝕊$ we have $H(x,0)=x_0$ (for some $x_0\in 𝕊$) and $H(x,1)=x$. Define a map $r:𝔹\to 𝕊$ as follows:

It is easy to see that this formula defines a retraction from $𝔹$ onto $𝕊$. Contradiction. $\mathrm{\square }$


Note that this proposition does not state that any of the conditions $(1),(2),(3)$ hold. It only states that they are equivalent. It is well known that all of them are true if $V$ is finite dimensional and all are false if $V$ is infinite dimensional.