Review of Precompact Sets and Compact Sets
Review of Precompact Sets and Compact Sets
Let $E$ be a locally convex topological vector space and let $A \subseteq E$.
- Recall from the Precompact Sets in a LCTVS page that if $U$ is a neighbourhood of the origin then $A$ is said to be Small of Order $U$ if for all $x, y \in A$ we have that $x - y \in U$.
- We said that $A$ is Precompact if for every absolutely convex neighbourhood $U$ of the origin there exists a finite collection $A_1$, $A_2$, …, $A_n$ of sets that are all small of order $U$ and such that:
\begin{align} \quad A \subseteq \bigcup_{i=1}^{n} A_i \end{align}
- We summarize some equivalent criterion for precompactness of a set $A$ in a locally convex topological vector space.
Equivalent Criteria for Precompactness | |
(1) | $A$ is precompact. |
(2) | For every absolutely convex neighbourhood $U$ of the origin there exists $a_1, a_2, ..., a_n \in E$ such that $\displaystyle{A \subseteq \bigcup_{i=1}^{n} (a_i + U)}$. |
- On the Basic Properties of Precompact Sets in a LCTVS page we looked at some properties of precompact sets, which are summarized below.
Properties of Precompact Sets | |
(1) | If $A$ is precompact then $\overline{A}$ is precompact. |
(2) | If $A$ is precompact then for all $\lambda \in \mathbf{F}$, $\lambda A$ is precompact. |
(3) | If $A$ is precompact and $B \subseteq A$ then $B$ is precompact. |
(4) | Union of finite collections of precompact sets are precompact. |
(5) | Intersections of arbitrary collections of precompact sets are precompact. |
(6) | Sums of finite collections of precompact sets are precompact. |
- On the Boundedness of Precompact Sets in a LCTVS page we proved that:
If $E$ is a locally convex topological vector space then every precompact set is a bounded set.
- On the For Hausdorff LCTVS, Precompact Neighbourhood Implies Finite-Dimensional we proved the following useful result:
If $E$ is a Hausdorff locally convex topological vector space and if $E$ has a precompact neighbourhood of the origin then $E$ is finite-dimensional.
- On the Results Regarding Precompactness we also noted some useful results regarding precompactness which are summarized below.
Results on Precompactness | |
(1) | If $(E, F)$ is a dual pair and $A$ is $\sigma(E, F)$-weakly bounded then $A$ is weakly precompact. |
(2) | If $A$ is precompact then $\mathrm{conv}(A)$ is precompact. |
(3) | If $A$ is precompact then $\mathrm{abs \: conv}(A)$ is precompact. |
(4) | If $A$ is precompact then $\overline{\mathrm{abs \: conv}(A)}$ is precompact. |
- On the Compact Sets in a Topological Space page we defined $A$ to be Compact if for every open cover of $A$ there is a finite subcollection of the cover that still covers $A$.
- We summarize some results regarding compactness from the page above and from Basic Properties of Compact Sets in a LCTVS below.
Properties of Compactness | |
(1) | Unions of finite collections of compact sets are compact. |
(2) | Closed subsets of compact sets are compact. |
(3) | If $E$ is Hausdorff and $K$ is compact then $K$ is closed. |
(4) | Continuous images of compact sets are compact. |
(5) | If $K$ is compact then $\lambda K$ is compact for each $\lambda \in \mathbf{F}$. |
(6) | Finite sums of compact sets are compact. |
(7) | If $K$ is compact and $C$ is closed then $K + C$ is closed. |