Table of Contents

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:
 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 FiniteDimensional if it has a base that is a finiteset, and InfiniteDimensional 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:
 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:
 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:
 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:
 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)  $p(\lambda x) = \lambdap(x)$ 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:
 We noted that if $A$ is absolutely convex and absorbent then $p_A$ is a seminorm on $E$ and that:
 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)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.