Review of Topological Vector Spaces

• Recall from the Topological Vector Spaces over the Field of Real or Complex Numbers page that a Topological Vector Space is a vector space with a topology for which the operations of addition and scalar multiplication are continuous. We state some useful facts about a topological vector space $E$ below.

	Results on Topological Vector Spaces
(1)	For each $a \in E$, $x \mapsto a + x$ is a homeomorphism of $E$ onto $E$.
(2)	For each $\alpha \in \mathbf{F}$, $\lambda \neq 0$, $x \mapsto \alpha x$ is a homeomorphism of $E$ onto $E$.

• On the Bases of Neighbourhoods for a Point in a Topological Vector Space we looked at bases of neighbourhoods of points in a topological vector space. Recall that if $E$ is a topological vector space and if $\mathcal U$ is a base of neighbourhoods of the origin then $\mathcal U + a$ is a base of neighbourhoods of the point $a$. If $\mathcal U$ is a base of neighbourhoods of the origin then the following properties are satisfied:

	Properties of a Base of Neighbourhoods of the Origin for a Topological Vector Space
(1)	Every $U \in \mathcal U$ is an absorbent set.
(2)	For each $U \in \mathcal U$ there exists a balanced $V \in \mathcal U$ with $V \subseteq V + V \subseteq U$.
(3)	For every convex $U \in \mathcal U$ there is an absolutely convex $W \in \mathcal U$ with $W \subseteq U$.

• We then looked at some useful results regarding closures of convex, balanced, and absolutely convex sets in a topological vector spaces on the The Closure of a Convex Set in a TVS, The Closure of a Balanced Set in a TVS, and The Closure of an Absolutely Convex Set in a TVS pages which are summarized below.

	Closures in a Topological Vector Space
(1)	If $E$ is a topological vector space and $A \subseteq E$ is convex then $\overline{A}$ is convex.
(2)	If $E$ is a topological vector space and $A \subseteq E$ is balanced then $\overline{A}$ is balanced.
(3)	If $E$ is a topological vector space and $A \subseteq E$ is absolutely convex then $\overline{A}$ is absolutely convex.

• On the Every TVS Has a Base of Closed and Balanced Neighbourhoods of the Origin page we learned a special property of topological vector spaces:

If $E$ is a topological vector space then $E$ has a base of closed and balanced neighbourhoods of the origin.

• On the Criterion for a Topological Vector Space to be Hausdorff we proved an important criterion for a topological vector space to be Hausdorff:

A topological vector space $E$ is Hausdorff if and only if $\displaystyle{\bigcap_{U \in \mathcal U} U = \{ o \}}$ where $\mathcal U$ is a base of neighbourhoods of the origin.

• On the Locally Convex Topological Vector Spaces over the Field of Real or Complex Numbers we defined a Locally Convex Topological Vector Space to be a topological vector space that has a base of convex neighbourhoods of the origin (and thus each point has a base of convex neighbourhoods).

• On the Every LCTVS Has a Base of Closed Absolutely Convex Absorbent Neighbourhoods of the Origin page we have that:

If $E$ is a locally convex topological vector space then $E$ has a base of closed absolutely convex and absorbent neighbourhoods $\mathcal U$ of the origin with the property that if $U, V \in \mathcal U$ then there exists a $W \in \mathcal U$ with $W \subseteq U \cap V$ and such that if $U \in \mathcal U$ and $\alpha \in \mathbf{F}$ with $\alpha \neq 0$ then $\alpha U \in \mathcal U$.