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:

(1)

\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 $\epsilon > 0$.