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.
Solution.
From Exercise 1.1.7 we know that \(\n f_{C^0([0,1])} =
\n f_{\sup} = 1+e\text{.}\) Next we calculate
\begin{align*}
\n{f'}_{C^0([0,1])}
\amp =
\sup_{t \in [0,1]} \abs{f'(t)}\\
\amp =
\sup_{t \in [0,1]} \abs{1+e^t}\\
\amp =
1+e\text{,}
\end{align*}
where we have used that \(t \mapsto 1+e^t\) is positive an increasing on \([0,1]\text{.}\) Similarly we find
\begin{align*}
\n{f''}_{C^0([0,1])}
\amp =
\sup_{t \in [0,1]} \abs{f''(t)}\\
\amp =
\sup_{t \in [0,1]} \abs{e^t}\\
\amp =
e\text{.}
\end{align*}
Putting everything together, we have
\begin{align*}
\n f_{C^2([0,1])}
\amp =
\n f_{C^0([0,1])}
+ \n{f'}_{C^0([0,1])}
+ \n{f''}_{C^0([0,1])}\\
\amp =
(1+e)+(1+e)+e\\
\amp =
2+3e\text{.}
\end{align*}
Comment.
Many students wrote, e.g., \(\n{e^t}_{C^0}\) as a shorthand for the \(C^0([0,1])\) norm of the function \(t \mapsto e^t\text{.}\) This breaks our rules laid down in Section 1.6, because it conflates the expression (or maybe formula) \(e^t\) with the function \(t \mapsto e^t\text{,}\) \([0,1] \to \R\text{.}\) So technically it would be more correct to write something like \(\n{t\mapsto
e^t}_{C^0}\text{.}\) In the context of this relatively simple exercise, though, this didn’t seem to cause any real confusion.
On an exam I would not take off marks for this particular abuse of notation in and of itself. But do be warned that in more complicated problems this type of notational abuse can lead to real confusion and many lost marks. Be careful!
(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.
Solution.
From Exercise 1.1.8 we know that \(\n f_{C^0([-1,2])} = \n
f_{\sup} \le 11\text{.}\) Estimating each term in the sum separately, we similarly find
\begin{align*}
\n{f'}_{C^0([-1,2])}
\amp= \sup_{t \in [-1,2]}
\abs{f'(t)}\\
\amp = \sup_{t \in [-1,2]}
\left|1+2t-\frac 32 t^2+ \frac 1{20}t^4\right|\\
\amp\le \sup_{t \in [-1,2]}
\left(1 + 2\abs t+\frac 32 \abs t^2+ \frac 1{20} \abs
t^4\right)\\
\amp\le
1 +
2\sup_{t \in [-1,2]} \abs t
+ \frac 32 \sup_{t \in [-1,2]} \abs t^2
+ \frac 1{20} \sup_{t \in [-1,2]} \abs t^4\\
\amp\le
1 +
2 \cdot 2
+ \frac 32 \cdot 2^2
+ \frac 1{20} \cdot 2^4\\
\amp = 11 + \frac 4{5}
\le 12\text{.}
\end{align*}
Thus
\begin{align*}
\n f_{C^1([-1,2])}
\amp =
\n f_{C^0([-1,2])}
+ \n{f'}_{C^0([-1,2])}\\
\amp \le 11 + 12 = 23\text{.}
\end{align*}
Comment 1.
See the first comment for the previous part of this exercise.
Comment 2.
Some students were very brief in their solutions to this part of the exercise, for instance writing
\begin{equation*}
\n{f'}_{C^0} \le 1 + 4 + 6 + \tfrac 5{20}
\end{equation*}
without any further explanation. On an exam this would certainly be living dangerously! If you happen to do the estimate exactly as I had expected you to, and get exactly the same sum on the right hand side, then I can at least guess at what your thought process might be. But if you do something else, e.g. if you write something confusing like
\begin{equation*}
\n{f'}_{C^0} \le 0 + 8 + 0 - \tfrac 1{20}\text{,}
\end{equation*}
then it becomes very hard for me to read your mind, and basically impossible for me to assign you even partial marks.
For this particular problem, I think the best thing to do is just give at least a few intermediate steps. Alternatively, you could try to explain in words what you are doing, e.g. “using the triangle inequality for \(\n\blank_{C^0}\) and estimating each term separately, I get…”.
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.
