Exercises 3.8 Exercises
1. \(C^0\) norm.
Prove that \(\|f\|_{C^0}=\sup_{x\in[0,1]}|f(x)|\) is a norm on \(C^k([0,1])\) for each \(k\geq0\text{.}\) What is the completion of \(C^k([0,1])\) with respect to \(\|\cdot\|_{C^0}\text{?}\)2. \(C^1\) norm.
Prove that \(\|f\|_{C^1}=\|f\|_{C^0}+\|f'\|_{C^0}\) is a norm on \(C^k([0,1])\) for each \(k\geq1\text{.}\) What is the completion of \(C^k([0,1])\) with respect to \(\|\cdot\|_{C^1}\text{?}\)3. A sequence.
Prove that the sequence \(f_n(x)=\sqrt{(x-0.5)^2+\frac{1}{n}}\) converges to \(f(x)=|x-0.5|\) in the \(C^0\) norm.4. \(L^1\) vs \(L^p\).
Show that, for \(p>1\text{,}\) \(L^p([0,1])\subset L^1([0,1])\) and that the inclusion is strict.5. Continuous almost everywhere.
Explain why saying that a \(f\in L^p([0,1])\) has a continuous representative is not equivalent to saying that \(f\) is continuous almost everywhere.6. Continuous injections from \(C^k\) to \(C^0\).
Prove that the natural injection \(i:C^k([0,1])\to C^0([0,1])\text{,}\) where \(i\) is substantially the identity map, is continuous for each \(k\geq0\text{.}\)7. Continuous injections from \(L^p\) to \(L^1\).
Prove that the natural injection \(i:L^p([0,1])\to L^1([0,1])\text{,}\) where \(i\) is substantially the identity map, is continuous for each \(p\geq1\text{.}\)8. Continuous injections from \(C^k\) to \(L^p\).
Prove that the natural injection \(i:C^k([0,1])\to L^p([0,1])\text{,}\) where \(i\) is substantially the identity map, is continuous for each \(k\geq0\) and \(p\geq1\text{.}\)9. \(L^p\) convergence 1.
Assume that \(f_n\in C^0([0,1])\) converges to \(f\) in the \(C^0\) topology. Show that \(f_n\to f\) in \(L^p\) for every \(p\in(0,\infty]\text{.}\)10. \(L^p\) convergence 2.
Let \(f_n\in L^\infty((0,1))\) be given by
\begin{equation*}
f_n(x)=\begin{cases}
n,\amp x\in(0,\frac{1}{n}]\\
0,\amp x\in(\frac{1}{n},1)\\
\end{cases}.
\end{equation*}
Show that, although \(f_n\to 0\) pointwise, \(\|f_n\|_{L^p}\to\infty\) for \(p>1\) and \(\|f_n\|_{L^1}=1\) for every \(n\text{,}\) so \(f_n\) does not converge to 0 in \(L^p\) for any \(p\text{.}\)11. \(L^p\) convergence 3.
For \(n\geq1\text{,}\) let \(k\) be the unique natural number such that \(2^k\leq n\leq2^{k+1}\) and set
\begin{equation*}
I_n=\left[\frac{n}{2^k}-1,\frac{n+1}{2^k}-1\right].
\end{equation*}
Let \(f_n=\chi_{I_n}\text{.}\) - Show that, for \(p\in[1,\infty)\text{,}\) \(\|f_n\|_{L^p}=2^{-k/p}\text{,}\) so that \(f_n\to0\) in \(L^p\text{.}\)
- Show that \(f_n(x)\in\bR\) does not converge for any \(x\in(0,1)\text{.}\)