Section 3.1 \(C^k([a,b])\)
We now study functions on \([a,b]\) that have derivatives.
Definition 3.1.
Let \(k \in \N\text{.}\) Then \(C^k([a,b])\) comprises all functions \(f \in C^0([a,b])\) such that \(f\) is \(k\) times continuously differentiable in \((a,b)\) and there exist \(g_1, \dotsc, g_k \in C^0([a,b])\) such that \(f^{(i)}(t) = g_i(t)\) for all \(t \in (a,b)\) and \(i = 1, \dotsc, k\text{.}\) For \(f \in C^k([a,b])\text{,}\) we define
\begin{equation*}
\n f_{C^k([a,b])} = \sum_{i = 0}^k \|g_i\|_{C^0([a,b])},
\end{equation*}
where \(g_0 = f\) and \(g_1, \dotsc, g_k \in C^0([a,b])\) are as above.
In other words, a function \(f \maps [a,b] \to \R\) belongs to \(C^k([a,b])\) if it is continuous on \([a,b]\) and \(k\) times continuously differentiable in \((a,b)\) and every derivative up to order \(k\) has a continuous extension to \([a,b]\text{.}\) It is common to abuse notation and write \(f^{(i)}\) for the continuous extension of the \(i\)-th derivative of \(f\) (rather than \(g_i\)). Then
\begin{equation*}
\n f_{C^k([a,b])} = \sum_{i = 0}^k \|f^{(i)}\|_{C^0([a,b])}.
\end{equation*}
Theorem 3.2.
For every \(k \in \N \cup \{0\}\text{,}\) the space \(C^k([a,b])\) is a Banach space.
Proof.
It is routine to check that \(C^k([a,b])\) is a vector space and \(\n\blank_{C^k([a,b])}\) is a norm, so we do not give the details here. It remains to prove that the space is complete.
We already know that this holds true for \(k = 0\) by Theorem 1.50.
Now we proceed by induction over \(k\text{.}\) Let \(k \ge 1\) and suppose that the statement is true for \(C^{k - 1}([a,b])\text{.}\) Consider a Cauchy sequence \((f_n)_{n \in \N}\) in \(C^k([a,b])\text{.}\) Then \((f_n)_{n \in \N}\) is a Cauchy sequence in \(C^{k - 1}([a,b])\) as well; hence it has a limit \(f = \lim_{n \to \infty} f_n\) in \(C^{k - 1}([a,b])\text{.}\) Moreover, the sequence \((f_n^{(k)})_{n \in \N}\) is a Cauchy sequence in \(C^0([a,b])\text{;}\) hence it has a limit \(g = \lim_{n \to \infty} f_n^{(k)}\) in \(C^0([a,b])\text{.}\)
We want to prove that \(f^{(k)} = g\) in \((a,b)\text{.}\) To this end, note that for any two points \(x, y \in [a,b]\text{,}\)
\begin{equation*}
f_n^{(k - 1)}(y) - f_n^{(k - 1)}(x) = \int_x^y f_n^{(k)}(t) \, dt
\end{equation*}
for all \(n \in \N\) by the fundamental theorem of calculus. By the above uniform convergence, this implies that
\begin{equation*}
\begin{split}
f^{(k - 1)}(y) - f^{(k - 1)}(x) \amp = \lim_{n \to \infty} (f_n^{(k - 1)}(y) - f_n^{(k - 1)}(x)) \\
\amp = \lim_{n \to \infty} \int_x^y f_n^{(k)}(t) \, dt \\
\amp = \int_x^y g(t) \, dt.
\end{split}
\end{equation*}
The fundamental theorem of calculus now implies that \(f^{(k)} = g\) in \((a,b)\text{.}\) Hence \(f \in C^k([a,b])\) and \(f = \lim_{n \to \infty} f_n\) in \(C^k([a,b])\text{.}\)
Exercises Exercises
1. (PS6) Calculating and estimating \(C^k\) norms.
(a)
Let \(f \maps [0,1] \to \R\) be the function \(f(t)=t+e^t\) from Exercise 1.1.7. Calculate the \(C^2\) norm \(\n f_{C^2([0,1])}\text{.}\)
Hint.
Half of the work is already done for you in Exercise 1.1.7.
(b)
Let \(f \maps [-1,2] \to \R\) be the function \(f(t)=t+t^2-t^3/2+t^5/100\) from Exercise 1.1.8. Find a constant \(C \gt 0\) such that \(\n f_{C^1([-1,2])} \le C\text{.}\)
Hint.
Half of the work is already done for you in Exercise 1.1.8.
2. Uniform continuity of derivatives.
Suppose that \(f \in C^0([a,b])\) is continuously differentiable on \((a,b)\text{,}\) and that \(f'\) is uniformly continuous on \((a,b)\text{.}\) Show that \(f \in C^1([a,b])\text{.}\)
Hint.
Lemma 2.4 may be useful.
