Normable Vector Spaces

Definition: Let $E$ be a topological vector space. Then $E$ is said to be Normable if there exists a norm $\| \cdot \|$ on $E$ such that the topology on $E$ coincides with the coarsest topology determined by $Q := \{ \| \cdot \| \}$, and the pair $(E, \| \cdot \|)$ is called a Normed Space.

Proposition 1: Let $E$ be a normed space. Then $E$ is metrizable.

• Proof: Let $\| \cdot \| : E \to [0, \infty)$ be the norm on $E$. Let $d : E \times E \to [0, \infty)$ be defined for all $x, y \in E$ by:

(1)

\begin{align} \quad d(x, y) := \| x - y \| \end{align}

• Then $d$ is a metric on $E$. Moreover, sets of the form $\{ x : \| x \| \leq \epsilon \}$ where $\epsilon > 0$ form a base of neighbourhoods of the origin. $\blacksquare$