Is the unit sphere of $C[0,1]$ connected?

Question

Let $C[0,1]$ be the space of all continuous functions on $[0,1]$.

The induced norm of $C[0,1]$ is supremum norm, i.e $$\|f\|= \sup_{x\in[0,1]} f(x) .$$ Under this norm, $C[0,1]$ is a Banach space. Moreover, it is a Banach algebra with the multiplication being usual multiplication.

My question is whether $S^1$, the unit sphere of $C[0,1]$, is connected or not.

Answer 1

It is path-connected, like the unit spheres of every normed (real or complex) space except $\Bbb R$: if $g\ne -f$, consider $\gamma(t)=\frac{tf+(1-t)g}{\lVert tf+(1-t)g\rVert_\infty}$. This is well defined because with these hypothesis $tf+(1-t)g=0$ for some $t\in[0,1]$ if and only if $g=-f$ and $t=\frac12$. If $g=-f$, make a path from $f$ to a third function $h$ (here you use $\dim_{\Bbb R} V>1$) and then one from $h$ to $g$.