Review of Bounded Sets in a LCTVS
Review of Bounded Sets in a LCTVS
- Recall from the Bounded Sets in a LCTVS page that if $E$ is a vector space and if $A$ and $B$ are subsets of $E$ then $A$ Absorbs $B$ if there exists a $\lambda > 0$ such that if $\mu in \mathbf{F}$ and $|\mu| \geq \lambda$ then:
\begin{align} \quad B \subseteq \mu A \end{align}
- If $E$ is a locally convex topological vector space then a subset $A$ of $E$ is said to be a Bounded Set if and only if any of the following hold:
Equivalent Criteria for Boundedness of a Set | |
Definition | For every neighbourhood $U$ of the origin, $A$ is absorbed by $U$. |
(1) | For every neighbourhood $U$ of the origin there exists a $\lambda_U > 0$ such that if $\mu \in \mathbf{F}$ and $|\mu| \geq \lambda_U$ then $A \subseteq \mu_U U$. |
(2) | For every absolutely convex neighbourhood $U$ there exists a $\lambda_U > 0$ such that $A \subseteq \lambda_U U$. |
(3) | $p(A)$ is a bounded set of nonnegative real numbers for every seminorm $p \in Q$, where $Q$ is a collection of seminorms on $A$ which determine the locally convex topology on $A$, i.e., sets of the form $\{ x : \sup_{1 \leq i \leq n} p_i(x) \leq \epsilon \}$ with $p_1, p_2, ..., p_n \in Q$ and $\epsilon > 0$ form a base of neighbourhoods of the origin. |
- On the The Closure, Convex Hull, and Absolutely Convex Hull of a Bounded Set is a Bounded Set in a LCTVS, Subsets, Scalar Multiples, Finite Unions, and Arbitrary Intersections of Bounded Sets in a LCTVS, and A Continuous Linear Image of a Bounded Set is a Bounded Set in a LCTVS pages we looked at some important properties of bounded sets in a locally convex topological vector space. These are summarized below:
Properties of Bounded Sets | |
Closures of Bounded Sets are Bounded | If $A$ is bounded then $\overline{A}$ is bounded. |
Convex Hulls of Bounded Sets are Bounded | If $A$ is bounded then $\mathrm{conv}(A)$ is bounded. |
Absolutely Convex Hulls of Bounded Sets are Bounded | If $A$ is bounded then $\mathrm{abs \: conv}(A)$ is bounded. |
Subsets of Bounded Sets are Bounded | If $A$ is bounded and $B \subseteq A$ then $B$ is bounded. |
Scalar Multiples of Bounded Sets are Bounded | If $A$ is bounded and $\lambda > 0$ then $\lambda A$ is bounded. |
Unions of Finitely Many Bounded Sets are Bounded | If $A_1$, $A_2$, …, $A_n$ are bounded then $\displaystyle{\bigcup_{i=1}^{n} A_i}$ is bounded. |
Intersections of Arbitrarily Many Bounded Sets are Bounded | If $\{ A_{\alpha} \}_{\alpha}$ is an arbitrary collection of bounded sets then $\displaystyle{\bigcap_{\alpha} A_{\alpha}}$ is bounded. |
Continuous Images of Bounded Sets are Bounded | If $A$ is bounded in $E$ and $t : E \to F$ is continuous then $t(A)$ is bounded in $F$. |
- On the Bounded Neighbourhood Criterion for a Hausdorff LCTVS to be Normable we looked at the following important criterion for a Hausdorff locally convex topological vector space to be normable:
If a Hausdorff locally convex topological vector space has a bounded neighbourhood of the origin then it is normable.
- Thus, a non-normable Hausdorff locally convex topological vector space is such that every neighbourhood of the origin is unbounded, and consequentially, for a metric space $(E, d)$, if $E$ is not normable then $B(o, \epsilon)$ is unbounded for every $\epsilon > 0$.