Examples of LCTVS - Spaces of Infinitely Differentiable Functions

1. Infinitely Differentiable Functions on $[a, b]$

Definition: The Space of Infinitely Differentiable Real-Valued or Complex-Valued Functions on $[a, b]$ is denoted by $C^{\infty}[a, b]$, and is equipped with the coarsest topology determined by the seminorms $Q := \{ p_n : n \in \{ 0 \} \cup \mathbb{N} \}$ where for each $n \in \mathbb{N}$, $\displaystyle{p_n(f) := \sup_{t \in [a, b]} |f^{(n)}(x)|}$.

A base of neighbourhoods of the origin is given by sets of the form:

(1)

\begin{align} \quad \{ f : \sup_{1 \leq i \leq n} p_i(f) \leq \epsilon \end{align}

for $n \in \mathbb{N}$ and for all $\epsilon > 0$.

Proposition 1: $C^{\infty}[a, b]$ equipped with the coarsest topology determined by the seminorms $Q := \{ p_n : n \in \{0 \} \cup \mathbb{N} \}$ is Hausdorff and has a countable base of neighbourhoods of the origin, and is thus metrizable.

• Proof: If $f \in C^{\infty}[a, b]$ is such that $f \neq 0$, then there exists an $x \in [a, b]$ such that $f(x) \neq 0$. So $p_0(f) > 0$. Thus $C^{\infty}[a, b]$ is Hausdorff.

• Meanwhile $\displaystyle{\left \{ f : \sup_{1 \leq i \leq n} p_i(f) \leq \frac{1}{m} \right \}_{n \in \mathbb{N} \cup 0, m \in \mathbb{N}}}$ is a countable base of the origin.

• Thus the locally convex topological vector space $C^{\infty}[a, b]$ is Hausdorff. $\blacksquare$

2. Infinitely Differentiable Functions on $(-\infty, \infty)$

Definition: The Space of Infinitely Differentiable Real-Valued or Complex-Valued Functions on $(-\infty, \infty)$ is denoted by $C^{\infty}(-\infty, \infty)$, and is equipped with the coarsest topology determined by the seminorms $Q := \{ p_{m, n} : m \in \mathbb{N}, n \in \{ 0 \} \cup \mathbb{N} \}$ where for each $n \in \mathbb{N}$, $\displaystyle{p_{m, n}(f) := \sup_{t \in [-m, m]} |f^{(n)}(x)|}$.

Proposition 2: $C^{\infty}(-\infty, \infty)$ equipped with the coarsest topology determined by the seminorms $Q := \{ p_{m, n} : m \in \mathbb{N}, n \in \{0 \} \cup \mathbb{N} \}$ is Hausdorff and has a countable base of neighbourhoods of the origin, and is thus metrizable.