Summary of continuous function spaces
In general differential calculus, we have learned the definitions of function continuity, such as functions of class \(C^0\) and \(C^2\). For most cases, we only take them for granted as for example, we have memorized the formulations of Green identities while ignored the conditions on function's continuities. Although this is helpful for a "vivid" and "naive" understanding, mathematical rigorousness and structural beauties are lost. Therefore, this article summarizes several definitions of function continuities and clarifies their imbedding relationships.
Definition (Continuous function space \(C^m(\Omega)\) with order \(m\)) For non-negative integer \(m\), \(C^m(\Omega)\) is the vector space consisting of all functions \(\phi\) with their partial derivatives \(D^{\alpha} \phi\) of orders \(0 \leq \abs{\alpha} \leq m\) continuous on an open set \(\Omega\).
Because the domain \(\Omega\) is open, functions in \(C^m(\Omega)\) may not be bounded, which seldom appear in engineering cases. Therefore, the spaces of bounded continuous functions \(C_B^m(\Omega)\) are introduced as below, which are subspaces of \(C^m(\Omega)\).
Definition (Bounded continuous function spaces \(C_B^m(\Omega)\)) \(C_B^m(\Omega)\) is a subspace of \(C^m(\Omega)\) and for all \(\phi \in C_B^m(\Omega)\), its partial derivatives \(D^{\alpha} \phi\) of orders \(0 \leq \abs{\alpha} \leq m\) are bounded on \(\Omega\). \(C_B^m(\Omega)\) is a Banach space with the norm defined as $$ \norm{\phi; C_B^m(\Omega)} = \max_{0 \leq \abs{\alpha} \leq m} \sup_{x \in \Omega} \abs{D^{\alpha} \phi(x)} $$
Because the domain \(\Omega\) is open in \(\mathbb{R}^n\), we have the intention to continuously extend the function to domain boundary \(\pdiff\Omega\) (note: This is very important because it determines for example whether our solution to a PDE can be extended continuously to match the given boundary condition). Then comes the definition of bounded and uniformly continuous function spaces \(C^{m}(\overline{\Omega})\).
Definition (Spaces of bounded and uniformly continuous functions \(C^{m}(\overline{\Omega})\)) \(C^{m}(\overline{\Omega})\) is a subspace of \(C_B^{m}(\Omega)\) and for all \(\phi \in C^{m}(\overline{\Omega})\), its partial derivatives \(D^{\alpha} \phi\) of orders \(0 \leq \abs{\alpha} \leq m\) are uniformly continuous on \(\Omega\). \(C^{m}(\overline{\Omega})\) is a Banach space with the norm defined as $$ \norm{\phi; C^m(\overline{\Omega})} = \max_{0 \leq \abs{\alpha} \leq m} \sup_{x \in \Omega} \abs{D^{\alpha} \phi(x)} $$
Remark
- A bounded and uniformly continuous function has a unique, bounded and continuous extension to \(\overline{\Omega}\).
- Uniform continuity can be understood as: a change in the function value anywhere in the function's range can control the change of independent variable uniformly, i.e. there is a global common bound on it. At the first glance, we tend to say that if a function is uniformly continuous, it is also bounded. However, this is not always true and we leave this question to future post.
Finally, we have the definition of Hölder and Lipschitz continuous function spaces.
Definition (Spaces of Hölder continuous functions \(C^{m,\lambda}(\overline{\Omega})\)) \(C^{m,\lambda}(\overline{\Omega})\) is a subspace of \(C^{m}(\overline{\Omega})\) and for all \(\phi \in C^{m,\lambda}(\overline{\Omega})\), its partial derivatives \(D^{\alpha} \phi\) satisfies a Hölder condition of exponent \(\lambda \in (0, 1]\), i.e. there exists a constant \(K\) such that
$$ \abs{D^{\alpha} \phi(x) - D^{\alpha} \phi(y)} \leq K\abs{x - y}^{\lambda} \quad (x, y \in \Omega) $$
When the Hölder exponent \(\lambda\) is 1, functions in the space \(C^{m,1}(\overline{\Omega})\) are Lipschitz continuous. \(C^{m,\lambda}(\overline{\Omega})\) is a Banach space with the norm defined as $$ \norm{\phi; C^{m,\lambda}(\overline{\Omega})} = \abs{\phi; C^m(\overline{\Omega})} + \max_{0 \leq \abs{\alpha} \leq m} \sup_{\overset{x, y \in \Omega}{x \neq y}} \frac{\abs{D^{\alpha} \phi(x) - D^{\alpha} \phi(y)}}{\abs{x - y}^{\lambda}} $$
Next, we'll show the imbedding chain of the above continuous function spaces.
Definition (Imbeddings) Let \(X\) and \(Y\) be normed spaces. \(X\) is imbedded in \(Y\), written as \(X \rightarrow Y\), if \(X\) is a vector subspace of \(Y\) and the identity operator \(I: X \longrightarrow Y\) is continuous for all \(x \in X\).
Remark The continuous identity operator implies that the norm of \(x \in X\) can be used to control the norm of \(y = I(x) \in Y\) with a constant \(M\),
$$ \norm{y; Y} \leq M \norm{x; X} \quad (\forall x \in X) $$
With the above defined norms for various continuous function spaces, the whole chain of imbeddings is summarized as follows when \(\Omega\) is convex:
\begin{align*} & C^{\infty} (\overline{\Omega}) \rightarrow \cdots C^{m+1}(\overline{\Omega}) \rightarrow C^{m,1}(\overline{\Omega}) \rightarrow C^{m,\lambda}(\overline\Omega) \\ & \rightarrow C^{m, v}(\overline\Omega) \rightarrow C^{m}(\overline\Omega) \rightarrow \cdots C^0(\overline{\Omega}) \end{align*}
where \(0 < v < \lambda < 1\).
