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:
(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.
 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 $F$.

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 $\epsilon > 0$.