Past exam of the functional analysis I course of the University of Oxford / 2023 / 1 / d / iii

The kernel $W=⟨1⟩$ is complete: the map $a\mapsto a1$ is an isometry from $\mathbb{R}$ with its absolute-value norm onto $W$ with the supremum norm.

Define a linear map $S:Y\to X$ byThe fundamental theorem of calculus gives $Sg\in C^1([-1,1])$ and $TSg=g$. Also,Thus $S$ is a bounded right inverse, and $Y=C([-1,1])$ is a Banach space. Applying part (c),