Examples Of LCTVS - Spaces of Continuous Functions

# Examples of LCTVS - Spaces of Continuous Real-Valued or Complex-Valued Functions

## 1. Continuous Real-Valued or Complex-Valued Functions on $[a, b]$

 Definition: The Space of Continuous Real-Valued or Complex-Valued Functions on $[a, b]$ is denoted by $C[a, b]$. This is a normed space with norm $\| \cdot \| : C[a, b] \to [0, \infty)$ defined for all $f \in C[a, b]$ by $\displaystyle{\| f \| := \sup_{t \in [a, b]} |f(t)|}$.

Since $[a, b]$ is compact, we are guaranteed that for each continuous function $f : [a, b] \to \mathbf{F}$, the supremum $\displaystyle{\sup_{t \in [a, b]} |f(t)|}$ exists.

Since $C[a, b]$ is a normed space, it is also a locally convex topological vector space.

## 2. Continuous Real-Valued or Complex-Valued Functions on $(-\infty, \infty)$

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

Indeed, for each $n \in \mathbb{N}$, $p_n : C(-\infty, \infty) \to [0, \infty)$ is a seminorm on $C(-\infty, \infty)$, since clearly $p_n(0) = 0$, $\displaystyle{p_n(\lambda f) = \sup_{t \in [-n, n]} |\lambda f(t)| = |\lambda| \sup_{t \in [-n, n]} |f(t)|}$, and $\displaystyle{p_n(f + g) = \sup_{t \in [-n, n]} |f(t) + g(t)| \leq \sup_{t \in [-n, n]} |f(t)| + \sup_{t \in [-n, n]} |g(t)|}$, for all $f, g \in C(-\infty, \infty)$ and for all $\lambda \in \mathbf{F}$. So the coarsest topology determined by $Q := \{ p_n : n \in \mathbb{N} \}$ makes $C(-\infty, \infty)$ a locally convex topological vector space (see the theorem on The Coarsest Topology Determined by a Set of Seminorms on a Vector Space page)

A base of neighbourhoods for $C(-\infty, \infty)$ with this topology, are sets of the form:

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

where $n \in \mathbb{N}$ and where $\epsilon \geq 0$. But since $p_m(f) \leq p_n(f)$ for each $m, n \in \mathbb{N}$ with $m \leq n$, we see that a base of neighbourhoods for $C(-\infty, \infty)$ with this topology is:

(2)
\begin{align} \quad \{ f : p_n(f) \leq \epsilon \} \end{align}

where $n \in \mathbb{N}$ and where $\epsilon \geq 0$. These neighbourhoods are the set of all continuous functions $f : (-\infty, \infty) \to \mathbf{F}$ that are bounded on $-n, n]$.

It should also be remarked that for each $n \in \mathbb{N}$, $p_n$ is not a norm on $C(-\infty, \infty)$. Indeed, take any continuous function $f : (-\infty, \infty) \to \mathbf{F}$ that vanishes on $[-n, n]$. Then $p_n(f) = 0$ but $f \neq 0$.

 Proposition 1: $C(-\infty, \infty)$ equipped with the coarsest topology determined by the set of seminorms $Q := \{ p_n : n \in \mathbb{N} \}$ is Hausdorff and has a countable base of neighbourhoods, thus it is metrizable.
• Proof: $C(-\infty, \infty)$ with the coarsest topology determined by the set of seminorms $Q$ is a locally convex topological vector space.
• Let $f \in C(-\infty, \infty)$ be a continuous function that is not identically the zero function. Then there exists an $x \in \mathbb{R}$ such that $f(x) \neq 0$. Take $N \in \mathbb{N}$ such that $|f(x)| < N$. Then $p_N(f) > 0$. Thus by the proposition on the Criterion for the Coarsest Topology Determined by a Set of Seminorms to be Hausdorff page we have that $C(-\infty, \infty)$ equipped with the coarsest topology determined by $Q = \{ p_n : n \in \mathbb{N} \}$ is Hausdorff.
• Furthermore, as mentioned above, $\{ f : p_n(f) \leq \epsilon \}_{n \in \mathbb{N}, \epsilon > 0}$ is a base of neighbourhoods of the origin, and furthermore, $\displaystyle{\left \{ f : p_n(f) \leq \frac{1}{m} \right \}_{m, n \in \mathbb{N}}}$ is a countable base of neighbourhoods of the origin.
• Thus, from the theorem on the Criterion for a LCTVS to be Metrizable page, we have that $C(-\infty, \infty)$ equipped with the coarsest topology determined by the seminorms $Q$ is metrizable.

## 3. Continuous Real-Valued or Complex-Valued Functions on a Compact Topological Space K

 Definition: The Space of Continuous Real-Valued or Complex-Valued Functions on a Compact Topological Space $K$ is denoted by $C(K)$. This is a normed space with norm $\| \cdot \| : K \to [0, \infty)$ defined for all $f \in C(K)$ by $\displaystyle{\| f \| := \sup_{t \in K} |f(t)|}$.

The compactness of $K$ guarantees us that for each continuous function $f : K \to \mathbf{F}$, the supremum $\displaystyle{\sup_{t \in K} |f(t)|}$ exists.

Again, since $C(K)$ is a normed space, it is also a locally convex topological vector space.

## 4. Continuous Real-Valued or Complex-Valued Functions on a Topological Space S

 Definition: The Space of Continuous Real-Valued or Complex-Valued Functions on a Topological Space $S$ is denoted by $C(S)$, and is equipped with the coarsest topology determined by the seminorms $Q := \{ p_A : A \subseteq S \: \mathrm{is \: compact} \}$ where for each compact subset $A \subseteq S$, $p_A(f) := \sup_{t \in A} |f(t)|$.

Again, $C(S)$ equipped with the coarsest topology determined by the seminorms $Q$ is a locally convex topological vector space.