= Solution
Put $M=2^{1/p'}$, the common <operator norm> from part (b). Given $\varepsilon>0$, choose $N$ with $\|f-f_N\|_p<\varepsilon/(3M)$. Since $f_N\in Y$, the scalar limit of $T_hf_N$ exists. The <Cauchy criterion for a function limit> gives $\delta>0$ such that
$$
|T_hf_N-T_{\widetilde h}f_N|<\frac\varepsilon3
\qquad(0<h,\widetilde h<\delta).
$$
Using the <triangle inequality> and the uniform <operator norm> bound,
$$
\begin{aligned}
|T_hf-T_{\widetilde h}f|
&\leq|T_h(f-f_N)|+|T_hf_N-T_{\widetilde h}f_N|+|T_{\widetilde h}(f_N-f)|\\
&\leq2M\|f-f_N\|_p+|T_hf_N-T_{\widetilde h}f_N|<\varepsilon.
\end{aligned}
$$
This is the requested estimate. The <completeness> of $\mathbb R$ makes the scalar <Cauchy criterion for a function limit> sufficient, hence $\lim_{h\to0^+}T_hf$ exists and $f\in Y$.
For $u,v\in Y$ and real $a,b$, the <linearity> of $T_h$ gives $T_h(au+bv)=aT_hu+bT_hv$, 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
$$
\boxed{Y\text{ is a closed linear subspace of }X.}
$$
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