topological properties of a given set

Question

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have used every sequence is $S$ has a convergent subsequence & to show $S$ connected I was wondering whether showing it path connected would help. However I am not sure.Is my approach correct? Please help

Answer 1

Note that , for $f(t)=|t-\frac12|$ we have $\int_0^1f=0$.

Hence for all $k\in \Bbb R$ , $\int_0^1kf=0$.

The set $\{kf\ , k\in \Bbb R \}$ is not bounded.

Answer 2

For any $a\in\mathbf{R}$ there is a line segment with $y=-a$ and $y=a$ on the endpoints of the interval $[0,1]$ (respectively) which integrates to $0$. So the inverse image of $0$ under $f\mapsto \int f$ on the interval is unbounded in the sup norm topology. For connectedness you should be able to create a path through symmetric functions integrating to $0$ from any one to any other: Scale one to the $0$ function, then scale the 0 function to the other.