Review of Seminorms and Norms

# Review of Seminorms and Norms

• Recall from the Seminorms and Norms on Vector Spaces page that if $E$ is a vector space then a Seminorm on $E$ is a function $p : E \to [0, \infty)$ that satisfies the following properties:
 Axioms of Seminorms (1) $p(x) \geq 0$ for all $x \in E$. (2) $p(\lambda x) = |\lambda|p(x)$ for all $x \in E$ and for all $\lambda \in \mathbf{F}$. (3) $p(x + y) \leq p(x) + p(y)$ for all $x, y \in E$.
• A Norm on $E$ is a seminorm on $E$ with the additional property that $p(x) = 0$ if and only if $x = o$. We summarize some basic properties of seminorms below.
 Basic Properties of Seminorms (1) If $p$ is a seminorm then $p^{-1}(0)$ is a subspace of $E$. (2) If $p$ is a seminorm then $|p(x) - p(y)| \leq p(x - y)$ for all $x, y \in E$. (3) If $p$ and $q$ are seminorms with the property that $q(x) \leq 1$ whenever $p(x) < 1$ then $q(x) \leq p(x)$ for all $x \in E$.
(1)
\begin{align} \quad p_A(x) := \{ \lambda : \lambda > 0 \: \mathrm{and} \: x \in \lambda A \} \end{align}
• We noted that if $A$ is absolutely convex and absorbent then $p_A$ is a seminorm on $E$ and that:
(2)
\begin{align} \quad \{ x : p_A(x) < 1 \} \subseteq A \subseteq \{ x : p_A(x) \leq 1 \} \end{align}
• More generally, if $p$ is a seminorm on $E$ then $\{ x : p(x) < \alpha \}$ and $\{ x : p(x) \leq \alpha \}$ are absolutely convex and absorbent subsets of $E$.
 Basic Properties of Gauges (1) If $\alpha \in \mathbf{F}$ and $\alpha \neq 0$ then $p_{\alpha A} = |\alpha|^{-1} p_A$. (2) If $A \subseteq B$ then $p_B(x) \leq p_A(x)$ for all $x \in E$. (3) $p_{A \cap B} = \sup \{ p_A, p_B \}$. (4) If $\{ x : p_A(x) < 1 \} \subseteq B \subseteq \{ x : p_A(x) \leq 1 \}$ then $p_A = p_B$.
• On the Continuity of Seminorms on Vector Spaces we investigated some properties of continuous seminorms. We noted that $p$ is continuous if and only if $p$ is continuous at the origin. When we look at gauges we have the following result:

If $A$ is absolutely convex and absorbent then $p_A$ is continuous if and only if $A$ is a neighbourhood of the origin, in which case $\mathrm{int}(A) = \{ x : p_A(x) < 1 \}$ and $\overline{A} = \{ x : p_A(x) \leq 1 \}$.

If $E$ is a vector space and $Q$ is a collection of seminorms on $E$ then $E$ equipped with the coarsest topology determined by $Q$ is a locally convex topological vector space. Furthermore, a base of neighbourhoods of the origin is given by sets of the form:

(3)
\begin{align} \quad \left \{ x : \sup_{1 \leq i \leq n} p_i(x) \leq \epsilon \right \} \end{align}

where $\epsilon > 0$ and $p_1, p_2, ..., p_n \in Q$.

• On the Normable Vector Spaces page we said that a topological vector space is Normable if there exists a norm $\| \cdot \|$ on $E$ such that the topology on $E$ is the coarsest topology determined by $\{ \| \cdot \| \}$. It is easy to see that if $E$ is normable then $E$ is metrizable with metric $d(x, y) := \| x - y \|$.

A locally convex topological vector space is metrizable if and only if its topology is Hausdorff and first countable.