Full Review of Topological Vector Spaces

1.1. Review of General Vector Spaces

• Recall from the Vector Spaces over the Field of Real or Complex Numbers page that a Vector Space is a set $E$ equipped with operations of addition and scalar multiplication which satisfy the following axioms:

	Axioms of a Vector Space
(1)	$x + y = y + x$ for all $x, y \in E$.
(2)	$x + (y + z) = (x + y) + z$ for all $x, y, z \in E$.
(3)	There exists a vector $o \in E$ such that $x + o = x$ for all $x \in E$.
(4)	For each $x \in E$ there exists a vector $-x \in E$ with $x + (-x) = o$.
(5)	$(\lambda \mu)x = \lambda (\mu x)$ for all $\lambda, \mu \in \mathbf{F}$ and for all $x \in E$.
(6)	$(\lambda + \mu)x = \lambda x + \mu x$ for all $\lambda, \mu \in \mathbf{F}$ and for all $x \in E$.
(7)	$\lambda(x + y) = \lambda x + \lambda y$ for all $\lambda \in \mathbf{F}$ and for all $x, y \in E$.
(8)	$1x = x$ for all $x \in E$.

• A Vector Subspace of $E$ is a subset $M \subseteq E$ such that $(x + y) \in M$ for all $x, y \in M$, and $\lambda x \in M$ for all $x \in M$ and for all $\lambda \in \mathbf{F}$. We noted that the set containing just the origin is always a vector subspace, and any intersection of subspaces is a subspace.

• On the Spanning Sets of Vectors page, if $A \subseteq E$, we defined a Linear Combination of points in $A$ to be a vector of the form:

(1)

\begin{align} \quad \lambda_1 x_1 + \lambda_2 x_2 + ... + \lambda_n x_n \end{align}

• with $x_1, x_2, ..., x_n \in A$ and $\lambda_1, \lambda_2, ..., \lambda_n \in \mathbf{F}$. The Span of $A$ denoted by $\mathrm{span}(A)$ is defined to be the set of all linear combinations of vectors in $A$.

• On the Linearly Independent Sets of Vectors page, we noted that if $A \subseteq E$ then $A$ is said to be Linearly Independent if for all $n \in \mathbb{N}$ and for all $x_1, x_2, ..., x_n \in A$ we have that the equation $\lambda_1 x_1 + \lambda_2x_2 + ... + \lambda_n x_n = o$ implies that $\lambda_1 = \lambda_2 = ... = \lambda_n = 0$.

• On the Bases for a Vector Space page we said that a Base for $E$ is a subset $A$ of $E$ that is linearly independent and spans $E$, and said that a vector space is Finite-Dimensional if it has a base that is a finite-set, and Infinite-Dimensional otherwise.

• We noted that every maximal linearly independent set is a base and every minimal spanning set is a base.

• On the Every Vector Space has a Base page we proved using the maximal axiom that every vector space has a base.

1.2. Review of Convex, Balanced, Absolutely Convex, and Absorbent Sets

Let $E$ be a vector space and let $A \subset E$.

Convex Sets

• Recall from the Convex and Balanced Sets of Vectors page that $A$ is Convex if for all $x, y \in A$ and for all $\lambda, \mu \in \mathbf{F}$ with $\lambda, \mu \geq 0$ and $\lambda + \mu = 1$ we have that:

(2)

\begin{align} \quad \lambda x + \mu y \in A \end{align}

• From the page mentioned above and the **Properties of Convex Sets of Vectors page, we summarize some properties of convex sets below.

	Properties of Convex Sets
(1)	If $A$ is convex and $x \in E$ then $x + A$ is convex.
(2)	If $A$ is convex and $\lambda \in \mathbf{F}$ then $\lambda A$ is convex.
(3)	If $A$ and $B$ are convex then $A + B$ is convex.
(4)	If $\{ A_i : i \in I \}$ is a collection of convex sets then $\displaystyle{\bigcap_{i \in I} A_i}$ is convex.
(5)	If $A$ is convex and $\lambda, \mu \in \mathbb{R}$ then $\lambda A + \mu A = (\lambda + \mu)A$.

Balanced Sets

• We said that $A$ is Balanced if for all $\lambda \in \mathbf{F}$ with $|\lambda| \leq 1$ we have that $\lambda A \subseteq A$.

• From the page mentioned at the beginning of the page, and from the **Properties of Balanced Sets of Vectors page, we summarize some properties of balanced sets below.

	Properties of Balanced Sets
(1)	If $A$ is a nonempty balanced set then $o \in A$.
(2)	If $\{ A_i : i \in I \}$ is a collection of balanced sets then $\displaystyle{\bigcup_{i \in I} A_i}$ is balanced.
(3)	If $\{ A_i : i \in I \}$ is a collection of balanced sets then $\displaystyle{\bigcap_{i \in I} A_i}$ is balanced.
(4)	If $A$ and $B$ are balanced sets then $A + B$ is a balanced set.
(5)	If $A$ is balanced and $\lambda \in \mathbf{F}$ then $\lambda A = |\lambda| A$.
(6)	If $A$ is balanced and $\lambda, \mu \in \mathbf{F}$ are such that $|\lambda| \leq |\mu|$ then $\lambda A \subseteq \mu A$.

Absolutely Convex Sets

• On the Absolutely Convex Sets of Vectors page we defined $A$ to be Absolutely Convex if it is both convex and balanced. We summarize some properties of absolutely convex sets below.

	Properties of Absolutely Convex Sets
(1)	If $A$ and $B$ are absolutely convex then $A + B$ is absolutely convex.
(2)	If $A$ is absolutely convex and $\lambda \in \mathbf{F}$ then $\lambda A$ is absolutely convex.
(3)	If $A$ is a nonempty absolutely convex set then $o \in A$.
(4)	If $A$ is absolutely convex and $\lambda, \mu \in \mathbf{F}$ are such that $|\lambda| \leq |\mu|$ then $\lambda A \subseteq \mu A$.

Absorbent Sets

• On the Absorbent Sets of Vectors page we defined $A$ to be Absorbent if for every $x \in E$ there exists a $\lambda > 0$ such that if $\mu \in \mathbf{F}$ is such that $|\mu| \geq \lambda$ then:

(3)

\begin{align} \quad x \in \mu A \end{align}

• From the page above and the Properties of Absorbent Sets of Vectors page we summarize some properties of absorbent sets below.

	Properties of Absorbent Sets
(1)	If $\{ A_1, A_2, ..., A_n \}$ is a finite collection of absorbent sets then $\displaystyle{\bigcap_{i=1}^{n} A_i}$ is absorbent.
(2)	If $A$ is absorbent then for each $x \in E$ there exists a $0 < \mu < 1$ such that $-\mu x \in A$.

1.3. Review of Topological Spaces

• Recall from the Topologies and Topological Spaces page that if $E$ is a set then a Topology on $E$ is a collection $\tau$ of subsets of $E$ called Open Sets which satisfy the following properties:

	Open Sets of a Topology
(1)	$\emptyset$ and $X$ are open sets..
(2)	The union of an arbitrary collection of open sets is an open set.
(3)	The intersection of a finite collection of open sets is an open set.

• The pair $(E, \tau)$ is called a Topological Space.

• The Closed Sets in a topological space are the complements of open sets. We have the following properties of closed sets:

	Closed Sets of a Topology
(1)	$\emptyset$ and $X$ are closed sets.
(2)	The union of an finite collection of closed sets is a closed set.
(3)	The intersection of an arbitrary collection of closed sets is a closed set.

• If $E$ is a topological space and $x \in E$ then a Neighbourhood of $x$ is a set $U$ such that there exists an open set $V$ with:

(4)

\begin{align} \quad x \in V \subseteq U \end{align}

• We summarize properties of neighbourhoods of points below.

	Properties of Neighbourhoods of Points
(1)	If $U$ is a neighbourhood of $x$ then $x \in U$.
(2)	If $U$ and $V$ are neighbourhoods of $x$ then $U \cap V$ is a neighbourhood of $x$.
(3)	If $U$ is a neighbourhood of $x$ and $U \subseteq V$ then $V$ is a neighbourhood of $x$.
(4)	If $U$ is a neighbourhood of $x$ then there exists a neighbourhood $V$ of $x$ such that $U$ is a neighbourhood of each $y \in V$.

• We noted that conversely, if $E$ is a set such that for each $x \in E$ there is a collection of sets $\mathcal U_x$ which satisfy properties (1)-(4) above, then there is a unique topology on $E$ such that for each $x \in E$, $\mathcal U_x$ is a set of neighbourhoods for $x$.

• We said that a topological space $E$ is Hausdorff if for every pair of distinct points $x, y \in E$ there exists open sets $U$ and $V$ with $x \in U$, $y \in V$, and $U \cap V = \emptyset$.

• On The Interior and Closure of a Set of Points page we defined the Interior of $A$, denoted by $\mathrm{int}(A)$, as the set of all Interior Points of $A$, i.e., the set of all $x \in A$ for which there exists an open set $U$ with $x \in U \subseteq A$. We proved that $A$ is open if and only if $A = \mathrm{int}(A)$.

• We defined the Closure of $A$, denoted by $\overline{A}$, as the set of all Points of Closure of $A$, i.e., the set of all $x \in E$ for which $U \cap A \neq \emptyset$ for all neighbourhoods $U$ of $x$. We proved that $A$ is closed if and only if $A = \overline{A}$.

• On the Bases of Neighbourhoods for a Point page we defined a Base of Neighbourhoods of $x$ to be a collection $\mathcal V_x$ of neighbourhoods of $x$ such that for every neighbourhood $U$ of $x$ there is a $V \in \mathcal V_x$ with $x \in V \subseteq U$. We noted that the collection of all open neighbourhoods of $x$ is a base of neighbourhoods of $x$.

• On the The Induced Topology on a Subset of a Topological Space we said that if $A \subseteq E$ then the Induced Topology on $A$ is the topology $\{ A \cap U : U \in \tau \}$. That is, every open set in $A$ is the intersection of $A$ with an open set in $E$. We then observed that every closed set in $A$ is the intersection of $A$ with a closed set in $E$, and every neighbourhood of $x$ in $A$ is the intersection of $A$ with a neighbourhood of $x$ in $E$.

• We summarize some basic results regarding the induced topology.

	Properties of the Induced Topology
(1)	If $E$ is Hausdorff then $A$ with the induced topology is Hausdorff.
(2)	If $E$ is first countable then $A$ with the induced topology is first countable.
(3)	If $E$ is metrizable then $A$ with the induced topology is metrizable.

• On the Continuous Functions Between Topological Spaces page we said that if $E$ and $F$ are topological spaces then a function $f : E \to F$ is Continuous at $x$ if for every neighbourhood $V_{f(x)}$ of $f(x)$ there exists a neighbourhood $U_x$ of $x$ such that:

(5)

\begin{align} \quad f(U_x) \subseteq V_{f(x)} \end{align}

• We say that $f$ is **Continuous if it is continuous at each $x \in E$. We summarize some equivalent continuity criteria below.

	Continuity Criteria
(1)	$f$ is continuous.
(2)	For every open set $U$ in $F$, $f^{-1}(U)$ is an open set in $E$.
(3)	For every closed set $V$ in $F$, $f^{-1}(V)$ is a closed set in $E$.

• On the Homeomorphisms Between Topological Spaces page we said that a Homeomorphism between $E$ and $F$ is a bijective function $f : E \to F$ for which $f$ and $f^{-1}$ are continuous. We summarize some basic properties of homeomorphisms below.

	Properties of Homeomorphisms
**(1)	If $f : E \to F$ is a homeomorphism and $A \subseteq E$ then $f(\mathrm{int}(A)) = \mathrm{int}(f(A))$.
(2)	If $f : E \to F$ is a homeomorphism and $A \subseteq E$ then $f(\overline{A}) = \overline{f(A)}$.
(3)	$E$ is Hausdorff if and only if $F$ is Hausdorff.
(4)	$E$ is first countable if and only if $F$ is first countable.

• On the Metric Spaces and Metrizability page we defined a Metric Space to be a pair $(E, d)$ where $d : E \times E \to [0, \infty)$ has the following properties:

	Axioms of a Metric Space
(1)	$d(x, y) = 0$ if and only if $x = y$.
(2)	$d(x, y) = d(y, x)$ for all $x, y \in E$.
(3)	$d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in E$.

• It should be noted that every metric space is a topological space. Furthermore, every metric space is Hausdorff and first countable.

1.4. 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 $\alpha U \in \mathcal U$.

1.5. Review of Seminorms and Norms

• Recall from the Seminorms and Norms on Vector Spaces page that if $E$ is a vector space then a Seminorm on $E$ is a function $p : E \to [0, \infty)$ that satisfies the following properties:

	Axioms of Seminorms
(1)	$p(x) \geq 0$ for all $x \in E$.
(2)	$p(\lambda x) = |\lambda|p(x)$ for all $x \in E$ and for all $\lambda \in \mathbf{F}$.
(3)	$p(x + y) \leq p(x) + p(y)$ for all $x, y \in E$.

• A Norm on $E$ is a seminorm on $E$ with the additional property that $p(x) = 0$ if and only if $x = o$. We summarize some basic properties of seminorms below.

	Basic Properties of Seminorms
(1)	If $p$ is a seminorm then $p^{-1}(0)$ is a subspace of $E$.
(2)	If $p$ is a seminorm then $|p(x) - p(y)| \leq p(x - y)$ for all $x, y \in E$.
(3)	If $p$ and $q$ are seminorms with the property that $q(x) \leq 1$ whenever $p(x) < 1$ then $q(x) \leq p(x)$ for all $x \in E$.

• On The Gauge of an Absolutely Convex and Absorbent Set that if $A$ is an absolutely convex and absorbent set in a vector space, then the Gauge of $A$ is the function $p_A : E \to [0, \infty)$ defined for all $x \in E$ by:

(6)

\begin{align} \quad p_A(x) := \{ \lambda : \lambda > 0 \: \mathrm{and} \: x \in \lambda A \} \end{align}

• We noted that if $A$ is absolutely convex and absorbent then $p_A$ is a seminorm on $E$ and that:

(7)

\begin{align} \quad \{ x : p_A(x) < 1 \} \subseteq A \subseteq \{ x : p_A(x) \leq 1 \} \end{align}

• More generally, if $p$ is a seminorm on $E$ then $\{ x : p(x) < \alpha \}$ and $\{ x : p(x) \leq \alpha \}$ are absolutely convex and absorbent subsets of $E$.

• On the Properties of Gauges of Absolutely Convex and Absorbent Sets page we looked at some basic properties of gauges of absolutely convex and absorbent sets $A$ and $B$ which are summarized below.

	Basic Properties of Gauges
(1)	If $\alpha \in \mathbf{F}$ and $\alpha \neq 0$ then $p_{\alpha A} = |\alpha|^{-1} p_A$.
(2)	If $A \subseteq B$ then $p_B(x) \leq p_A(x)$ for all $x \in E$.
(3)	$p_{A \cap B} = \sup \{ p_A, p_B \}$.
(4)	If $\{ x : p_A(x) < 1 \} \subseteq B \subseteq \{ x : p_A(x) \leq 1 \}$ then $p_A = p_B$.

• On the Continuity of Seminorms on Vector Spaces we investigated some properties of continuous seminorms. We noted that $p$ is continuous if and only if $p$ is continuous at the origin. When we look at gauges we have the following result:

If $A$ is absolutely convex and absorbent then $p_A$ is continuous if and only if $A$ is a neighbourhood of the origin, in which case $\mathrm{int}(A) = \{ x : p_A(x) < 1 \}$ and $\overline{A} = \{ x : p_A(x) \leq 1 \}$.

• On The Coarsest Topology Determined by a Set of Seminorms on a Vector Space page we defined the Coarsest Topology Determined by $Q$ (where $Q$ is a collection of seminorms on $E$) to be the coarsest topology for which every seminorm $Q$ is continuous with respect to the topology. We then looked at the important theorem.

If $E$ is a vector space and $Q$ is a collection of seminorms on $E$ then $E$ equipped with the coarsest topology determined by $Q$ is a locally convex topological vector space. Furthermore, a base of neighbourhoods of the origin is given by sets of the form:

(8)

\begin{align} \quad \left \{ x : \sup_{1 \leq i \leq n} p_i(x) \leq \epsilon \right \} \end{align}

where $\epsilon > 0$ and $p_1, p_2, ..., p_n \in Q$.

• On the Criterion for the Coarsest Topology Determined by a Set of Seminorms to be Hausdorff page we proved that if $E$ is equipped with the coarsest topology determined by a collection of seminorms $Q$, then $E$ is Hausdorff if and only if for every nonzero $x \in E$ there exists a seminorm $p \in Q$ such that $p(x) > 0$.

• On the Normable Vector Spaces page we said that a topological vector space is Normable if there exists a norm $\| \cdot \|$ on $E$ such that the topology on $E$ is the coarsest topology determined by $\{ \| \cdot \| \}$. It is easy to see that if $E$ is normable then $E$ is metrizable with metric $d(x, y) := \| x - y \|$.

• On the Criterion for a LCTVS to be Metrizable page we looked at a criterion for a locally convex topological vector space to be metrizable.

A locally convex topological vector space is metrizable if and only if its topology is Hausdorff and first countable.