Table of Contents

Examples of LCTVS  Spaces of Continuous RealValued or ComplexValued Functions
1. Continuous RealValued or ComplexValued Functions on $[a, b]$
Definition: The Space of Continuous RealValued or ComplexValued 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 RealValued or ComplexValued Functions on $(\infty, \infty)$
Definition: The Space of Continuous RealValued or ComplexValued 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)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)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 RealValued or ComplexValued Functions on a Compact Topological Space K
Definition: The Space of Continuous RealValued or ComplexValued 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 RealValued or ComplexValued Functions on a Topological Space S
Definition: The Space of Continuous RealValued or ComplexValued 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.