For a compact metric space $X$, $C(X)$ denotes the space of continuous real-valued functions on $X$ equipped with the supremum norm. Let $X$ and $Y$ be compact metric space and let $g:X \to Y$ be a continuous map. Define $T: C(Y) \to C(X)$ by $T(f) = f\circ g$ . Clearly, $T$ is a linear transformation.
- Prove that $T$ is bounded. What is the value of $\|T\|$?
- Give a necessary and sufficient condition on $g$ for $T$ to be onto.
- Give a necessary and sufficient condition on $g$ for $T$ to be one-to-one.
- Give a necessary and sufficient condition on $g$ for $T$ to be an isometry.