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 $x \in E$. |
- On The Gauge of an Absolutely Convex and Absorbent Set that if $A$ is an absolutely convex and absorbent set in a vector space, then the Gauge of $A$ is the function $p_A : E \to [0, \infty)$ defined for all $x \in E$ by:
\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:
\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 $E$.
- On the Properties of Gauges of Absolutely Convex and Absorbent Sets page we looked at some basic properties of gauges of absolutely convex and absorbent sets $A$ and $B$ which are summarized below.
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 $\overline{A} = \{ x : p_A(x) \leq 1 \}$.
- On The Coarsest Topology Determined by a Set of Seminorms on a Vector Space page we defined the Coarsest Topology Determined by $Q$ (where $Q$ is a collection of seminorms on $E$) to be the coarsest topology for which every seminorm $Q$ is continuous with respect to the topology. We then looked at the important theorem.
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 Criterion for the Coarsest Topology Determined by a Set of Seminorms to be Hausdorff page we proved that if $E$ is equipped with the coarsest topology determined by a collection of seminorms $Q$, then $E$ is Hausdorff if and only if for every nonzero $x \in E$ there exists a seminorm $p \in Q$ such that $p(x) > 0$.
- 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 $d(x, y) := \| x - y \|$.
- On the Criterion for a LCTVS to be Metrizable page we looked at a criterion for a locally convex topological vector space to be metrizable.
A locally convex topological vector space is metrizable if and only if its topology is Hausdorff and first countable.