Various Function Spaces 📂Hilbert Space

Various Function Spaces

Definition

A set of functions $X$ is called a function space if it forms a vector space.

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 $L^{p}$
$L^{p} (E) : = \left\{ f : \int_{E} | f |^{p} dm < \infty \right\}$
- Weighted Lebesgue space
Space of convergent sequences $\ell^{p}$
Sobolev space $W^{m,\ p}$
$W^{m,\ p}(\Omega):=\left\{ u \in L^p(\Omega)\ :\ D^\alpha u \in L^p(\Omega),\ 0\le |\alpha | \le m \right\}$