Various Function Spaces
Definition
A set of functions $X$ is called a function space if it forms a vector space.
Explanation
In the function space $X$, the inner product is defined by integration as follows.
$$ \langle f, g \rangle = \int f(x) g(x) dx,\quad f,g\in X $$
The main function spaces considered include the following.
Space of continuous functions $C^{m}$
$$ C^{m}(\mathbb{R}) : =\left\{ f \in C(\mathbb{R}) : f^{(n)} \text{ is continuous } \forall n \le m \right\} $$
- Space of test functions $C_{c}^{\infty} = \mathcal{D}$
- Schwartz space $\mathcal{S}$
- Hölder continuous function space
Lebesgue space $L^{p}$
$$ L^{p} (E) : = \left\{ f : \int_{E} | f |^{p} dm < \infty \right\} $$
Sobolev space $W^{m,\ p}$
$$ W^{m,\ p}(\Omega):=\left\{ u \in L^p(\Omega)\ :\ D^\alpha u \in L^p(\Omega),\ 0\le |\alpha | \le m \right\} $$