Section 2.4 Regularity of functions
In this short section we introduce some terminology about the regularity of functions, and recall some results on function spaces from MA30252.
Notation 2.19. Regularity of functions.
- For a nonempty subset \(A\) of \(\R^N\) (not necessarily open), \(C^0(A)\) is the space of continuous functions \(A \to \R\text{.}\)
- Letting \(\Omega\) denote a nonempty open subset of \(\R^N\) (as it always will!), \(C^k(\Omega)\) is the space of functions \(f \maps \Omega \to \R\) whose partial derivatives \(\partial^\alpha f\) of order \(\abs \alpha \le k\) exist and lie in \(C^0(\Omega)\text{.}\)
- We say \(f \in C^k(\overline \Omega)\) if \(f \in C^k(\Omega)\) and each partial \(\partial^\alpha f\) of order \(\abs \alpha \le k\) can be extended to an element of \(C^0(\overline \Omega)\text{.}\)
- A function \(f \in C^\infty(\Omega)\) if it is \(C^k(\Omega)\) for all \(k \ge 0\text{.}\)
- \(C^k(\Omega,\R^M)\) is the space of functions \(F \maps \Omega \to \R^M\) such that \(F_i \in C^k(\Omega)\) for \(i=1,\ldots,M\text{.}\) The spaces \(C^k(\overline\Omega,\R^M)\text{,}\) \(C^k(\Omega,\R^{M \times K})\) and \(C^k(\overline \Omega,\R^{M \times K})\) are defined similarly.
Of all of these spaces of functions, the spaces \(C^k(\overline\Omega)\) are perhaps the most amenable to analysis, since they are Banach spaces.
Theorem 2.20. \(C^k(\overline\Omega)\) is Banach.
Let \(\Omega \subset \R^N\) be a bounded open set and let \(k \in \N\text{.}\) Then \(C^k(\overline\Omega)\) is a Banach space with the norm
1
Following common practice, in these notes I will not put a bar over \(\Omega\) in the subscript for the norm. But I will not be picky about it if you prefer to add the bar.
\begin{equation*}
\n f_{C^k(\Omega)} = \sum_{\abs\alpha \le k} \sup_\Omega \abs{\partial^\alpha f}\text{,}
\end{equation*}
Proof.
See MA30252 for the case when \(N=1\text{.}\) The proof for general \(N\) is similar.
Remark 2.21.
We will apply Theorem 2.20 in the following way. Suppose that \(\Omega \subset \R^N\) is bounded, and \(f_n\) is a sequence in \(C^k(\overline\Omega)\) which is Cauchy. That is, suppose that for all \(\varepsilon \gt 0\) there exists \(M \in \N\) such that \(\n{f_n-f_m}_{C^k(\Omega)} \lt
\varepsilon\) for all \(n,m \ge M\text{.}\) Then Theorem 2.20 implies that this sequence is convergent in the sense that there exists a (unique) limit \(f \in C^k(\overline\Omega)\) such that \(\n{f_n-f}_{C^k(\overline\Omega)} \to
0\) as \(n \to \infty\text{.}\)
In Chapter 4, we will also use the following classic result on polynomial approximation.
Theorem 2.22. Weierstrass approximation theorem.
Let \(K \subset \R^N\) be compact. Then the set of polynomials is dense in \(C^0(K)\text{.}\) That is, for every \(f \in C^0(K)\) there exists a sequence of polynomials \(p_n\) with \(\n{p_n-f}_{C^0(K)} \to 0\) as \(n \to \infty\text{.}\)
Proof.
See MA30252.
Exercises Exercises
1. Series of \(C^k\) functions.
Let \(k \in \N\) and let \(\Omega\) be bounded. Suppose \(f_n\) is a sequence in \(C^k(\overline\Omega)\) and the infinite series
\begin{equation*}
\sum_{n=1}^\infty \n{f_n}_{C^k(\Omega)}
\end{equation*}
converges.
(a)
Show that the sequence of partial sums \(\sum_{i=1}^n f_i\) is Cauchy in \(C^k(\overline\Omega)\text{.}\)
(b)
Conclude from Theorem 2.20 that the infinite series \(\sum_{n=1}^\infty f_n \) converges in \(C^k(\overline\Omega)\text{.}\)
(c)
Set \(N=2\) and \(\Omega = (0,2\pi)^2\text{.}\) Use the previous parts to show that the infinite series
\begin{equation*}
\sum_{n=1}^\infty \frac{\cos(nx_1) \sin(nx_2)}{n^3}
\end{equation*}
defines a function in \(C^1(\overline\Omega)\text{.}\)
