Let $X=C[0,1]$ be the space of continuous functions on the interval $[0,1]$, $t_0\in J$. I am trying to show that if $T: X\to X$ defined by $Tx=vx$, where $v\in X$ is fixed, and $v(t_0)=0$, then $T$ is not invertible.
Here is the claim that I saw: Let $y=Tx=vx$. Then
(claim): $T^{-1}$ exists if and only if $x=y/v$ for all $t\in [0,1]$.
But because $y/v$ is not defined at $t_0$, T is not invertible. I don't understand why the claim is true. I see no reason why $y(t_0)=(vx)(t_0)=0$ can't hold without making our transformation, non-invertible.
Edit: The transformation is not surjective. My goodness.