Full Review of Topological Vector Spaces

# Full Review of Topological Vector Spaces

## 1.1. Review of General Vector Spaces

 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.

## 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}
 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 $\lambda A \subseteq A$.
 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}
 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

 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 $W \subseteq U$.
 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.

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

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.

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 $x \in E$.
(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 $E$.
 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 $\overline{A} = \{ x : p_A(x) \leq 1 \}$.

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 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 $d(x, y) := \| x - y \|$.

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