Past exam of the functional analysis I course of the University of Oxford / 2023 / 2 / c / Solution

Put $M=2^{1/p^{\prime }}$, the common operator norm from part (b). Given $\epsilon >0$, choose $N$ with $\Vert f-f_N{\Vert }_p<\epsilon /(3M)$. Since $f_N\in Y$, the scalar limit of $T_h{f}_N$ exists. The Cauchy criterion for a function limit gives $\delta >0$ such thatUsing the triangle inequality and the uniform operator norm bound,This is the requested estimate. The completeness of $\mathbb{R}$ makes the scalar Cauchy criterion for a function limit sufficient, hence ${\lim }_{h\to 0^{+}}{T}_{h}f$ exists and $f\in Y$.

For $u,v\in Y$ and real $a,b$, the linearity of $T_h$ gives $T_h(au+bv)=a{T}_{h}u+b{T}_{h}v$, which also has a limit. Thus $Y$ is a linear subspace. The preceding argument shows it contains every norm convergence limit of its sequences, so the sequential criterion for closed sets in a metric space gives