Full Review of Duality

# Full Review of Duality

## 2.1. Review of Linear Operators

• Recall from the Linear Operators Between Vector Spaces page that if $E$ and $F$ are vector spaces then a function $f : E \to F$ is a Linear Operator from $E$ to $F$ if for all $x, y \in E$ and for all $\lambda \in \mathbf{F}$ we have that $f(x + y) = f(x) + f(y)$ and $f(\lambda x) = \lambda f(x)$.
• When $E$ and $F$ are topological vector space, we have the following equivalences for when $f : E \to F$ is continuous.
 Criteria for Continuity in a Topological Vector Space (1) $f$ is continuous. (2) $f$ is continuous at the origin. (3) There exists a constant $M > 0$ such that $\| f(x) \| \leq M \| x \|$ for all $x \in E$. (This requires that $E$ and $F$ are normed spaces).
• On the Isomorphic Topological Vector Spaces we said that two topological vector spaces $E$ and $F$ are Isomorphic if there exists a bijective linear operator $f : E \to F$ for which both $f$ and $f^{-1}$ are continous.
• We then proved that if $E$ and $F$ are normed spaces and $f : E \to F$ is a linear bijection, then $f$ is an isomorphism if and only if there exists constants $M, N > 0$ such that for all $x \in E$:
(1)
\begin{align} \quad M \| x \| \leq \| f(x) \| \leq N \| x \| \end{align}

## 2.2. Review of Linear Forms and the Hahn-Banach Theorem

• We defined the Algebraic Dual to be $E^*$, the set of all linear forms on $E$, and proved the existence of many nonzero linear forms on $E$. In particular,

If $E$ is a vector space then for each $a \neq o$ there exists a $f \in E^*$ such that $f(a) \neq 0$.

• We also observed that if $E$ is a topological vector space then every $f \in E^*$ is an open map.
 Criteria for a Linear Form to be Continuous (1) $f$ is a continuous linear form. (2) There exists a neighbourhood $U$ of the origin for which $f(U)$ is a bounded set of real or complex numbers. (3) $f^{-1}(0)$ is a closed set (i.e., the null space is closed).
• We also proved the following basic properties regarding continuity when $E$ is further a locally convex topological vector space:
 Properties of Continuity in a Locally Convex Topological Vector Space (1) If $f$ is a linear form that is dominated by a continuous seminorm $p$ then $f$ is continuous. (2) If $f$ is a continuous linear form then $|f|$ is a continuous seminorm.
• On the Hyperplanes of a Vector Space page we defined a Hyperlane of a vector space $E$ to be a maximal and proper subspace $H$ of $E$, or equivalently, a subspace $H$ of $E$ for which $\mathrm{codim}(H) = 1$. We have the following characterizations of hyperplanes in topological vector spaces and locally convex topological vector spaces:
 Characiterizations of Hyperplanes in Topological Vector Spaces and Locally Convex Topological Vector Spaces (1) If $E$ is a topological vector space then $H$ is a hyperplane of $E$ if and only if there exists a nonzero linear form $f$ for which $f^{-1}(0) = H$. (2) If $E$ is a locally convex topological vector space then every hyperplane of $E$ is either closed or dense.
• We then turned our attention to The Hahn-Banach Theorem which is discussed on the following pages:
• The Hahn-Banach Theorem states:

(Hahn-Banach): If $E$ is a vector space, $M$ a subspace of $E$, $f$ a linear form on $M$, and $p$ a seminorm on $E$ for which $|f(x)| \leq p(x)$ for all $x \in M$, then there exists a linear form $f_1$ on $E$ which extends $f$ and such that $|f_1(x)| \leq p(x)$ for all $x \in E$.

• This we proven in multiple steps. We first proved:

If $E$ is a real locally convex topological vector space and if $H$ is subspace that does not intersect some open set $A$ in $E$, then either $H$ is a hyperplane OR there exists a point $x \not \in H$ for which $\mathrm{span} (H \cup \{ x \})$ still does not intersect $A$.

• We then proved that if $H$ is a real hyperplane of $E$, then $H \cap (iH)$ is a complex hyperplane of $E$. Then we showed that:

If $E$ is a locally convex topological vector and if $M$ is a subspace of $E$ that does not intersect some open and convex set $A$ then there must exist a closed hyperplane $H$ of $E$ that contains $M$ and that still does not intersect $A$.

• As a corollary, we observe that for a locally convex topological vector space $E$, a subspace $M$ is the intersection of all closed hyperplanes containing $M$, and from these results, we proved the Hahn-Banach theorem.
 Corollaries to the Hahn-Banach Theorem (1) If $E$ is a locally convex topological vector space and $M$ is a subspace of $E$ then every continuous linear form on $M$ can be extended to a continuous linear form on $E$. (2) If $E$ is a vector space then for each seminorm $p$ and each $a \in E$ there exists a linear form $f$ such that $|f(x)| \leq p(x)$ for all $x \in E$ and $f(a) = p(a)$. (3) If $E$ is a Hausdorff locally convex topological vector space and if $f(a) = 0$ for all $f \in E'$ then $a = o$.

(Hahn-Banach Separation): If $E$ is a locally convex topological vector space then every pair of disjoint convex sets $A$ and $B$ for which at least one of them is open, can be separated by a continuous linear form $f$, i.e., there exists an $f \in E'$ such that $f(A) \cap f(B) = \emptyset$.

• We summarize the corollaries to the Hahn-Banach separation theorem below.
 Corollaries to the Hahn-Banach Separation Theorem (1) If $E$ is a locally convex topological vector space and $B$ is convex with $a \not \in \overline{B}$ then there exists $f \in E'$ such that $f(a) \not \in \overline{f(B)}$. (2) If $E$ is a locally convex topological vector space and $B$ is absolutely convex with $a \not \in \overline{B}$ then there exists $f \in E'$ such that $|f(x)| \leq 1$ on $B$ and $f(a) > 1$. (3) If $E$ is a real locally convex topological vector space and $A$, $B$ is a pair of disjoint convex sets with $A$ open, then there exists $f \in E'$ and a constant $\alpha > 0$ such that $f(x) > \alpha$ on $A$ and $f(x) \leq \beta$ on $B$.

## 2.3. Review of Dual Pairs

• On the Dual Pairs of Vector Spaces page we defined a Dual Pair to be a triple $(E, F, \langle \cdot, \cdot \rangle)$ where $E$ and $F$ are vector spaces and $\langle \cdot, \cdot \rangle : E \times F \to \mathbf{F}$ is a bilinear form with the property that for each nonzero $x \in E$ there exists a $y \in F$ such that $\langle x, y \rangle \neq 0$ and with the property that for each nonzero $y \in F$ there exists a $x \in E$ such that $\langle x, y \rangle \neq 0$.
• We then observed some common examples of dual pairs:
 Examples of Dual Pairs Example 1: If $E$ is a vector space then $(E, E^*)$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E^*$ by $\langle x, f \rangle := f(x)$. Example 2: If $E$ is a Hausdorff locally convex topological vector space then $(E, E')$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E'$ by $\langle x, f \rangle := f(x)$. Example 3: If $(E, F, \langle \cdot, \cdot \rangle)$ is a dual pair then $(F, E, [\cdot, \cdot])$ with the duality pairing given for all $x \in E$ and for all $y \in F$ by $[y, x] := \langle x, y \rangle$.
• On The Weak Topology on E Determined by F page, for a dual pair $(E, F)$, we defined the Weak Topology on $E$ Determined by $F$ denoted by $\sigma(E, F)$, to be the coarsest topology determined by the collection of seminorms $Q := \{ p_y : y \in F \}$ where for each $y \in F$ we defined $p_y : E \to \mathbf{F}$ by:
(2)
\begin{align} \quad p_y(x) := |\langle x, y \rangle| \end{align}
• We also noted the following properties of the weak topology on $E$ determined by $F$, which are summarized below:
 Properties of the Weak Topology on $E$ Determined by $F$ (1) $\sigma(E, F)$ is the coarsest topology for which each of the seminorms $p_y$ are continuous. (2) $\sigma(E, F)$ is a Hausdorff locally convex topology. (3) A base of $\sigma(E, F)$-closed neighbourhoods of the origin are given by sets of the form $\displaystyle{\{ x : \sup_{1 \leq i \leq n} p_{y_i}(x) \leq 1 \}}$ with $y_1, y_2, ..., y_n \in F$.

If $(E, F)$ is a dual pair then the topological dual of $E$ equipped with $\sigma(E, F)$ is $F$, i.e.:

$(E^{\sigma(E, F)})' = F$

• On the Topologies of the Dual Pair (E, F) page we said that if $(E, F)$ is a dual pair then a Topology of the Dual Pair $(E, F)$ is a topology $\tau$ on $E$ that is locally convex with the property that $(E^{\tau})' = F$. Thus, the weak topology on $E$ determined by $F$ is a topology of the dual pair.
• We then proved a very important result regarding topologies of the dual pair $(E, F)$ and convex sets:

If $(E, F)$ is a dual pair and $A$ is a convex subset of $E$ then the closure of $A$ is the same subset when $E$ is equipped with any topology of the dual pair $(E, F)$.

## 2.4. Review of Polar and Bipolar Sets

Let $(E, F)$ be a dual pair and let $A \subseteq E$.

• On The Polar of a Set page we defined The Polar of $A$ in $F$ to be the subset $A^{\circ}$ which consists of all points $y \in F$ such that:
(3)
\begin{align} \quad \sup \{ |\langle x, y \rangle : x \in A \} \leq 1 \end{align}
• We then examined some useful properties of polars of sets which are summarized below:
 Properties of Polar Sets (1) If $A^{\circ}$ is absolutely convex and $\sigma(F, E)$-closed. (2) If $A \subseteq B$ then $B^{\circ} \subseteq A^{\circ}$. (3) For all $\lambda \in \mathbf{F}$ with $\lambda \neq 0$ then $(\lambda A)^{\circ} = |\lambda|^{-1} A^{\circ}$. (4) $\displaystyle{\left ( \bigcup_{\alpha} A_{\alpha} \right )^{\circ} = \bigcap_{\alpha} A_{\alpha}^{\circ}}$.
• On The Polar of a Subspace page we proved that if $(E, F)$ is a dual pair and $M$ is a subspace of $E$ then the polar $M^{\circ}$ in $F$ consists of all points $y \in F$ such that $\langle x, y \rangle = 0$ for all $x \in M$.
• In particular for the dual pair $(E, E^*)$, if $M$ is a subspace of $E$ then the polar $M^{\circ}$ in $E^*$ is $M^{\perp}$.

If $E$ is a Hausdorff locally convex topological vector space so that $(E, E')$ is a dual pair, then a subcollection $\mathcal V$ of continuous linear forms on $E$ is equicontinuous if and only if there exists a neighbourhood $U$ of the origin for which $\mathcal V \subseteq U^{\circ}$.

• On The Bipolar of a Set page said that if $(E, F)$ and $(F, G)$ are dual pairs and $A \subseteq E$ then the Bipolar of $A$ (in $G$) denoted by $A^{\circ \circ}$ is defined to be the polar (in $G$) of the polar of $A$ (in $F$).
• If $(E, F)$ and $(F, G)$ are dual pairs for which $E \subseteq G \subseteq F^*$ (as subspaces) then many useful properties can be deduced. These properties can be found on the following pages:
• These results are summarized below.
 Results Concerning Bipolars of Sets Characterization of Points in the Bipolar If $A \subseteq E$ then $z \in A^{\circ \circ}$ if and only if $|\langle z, y \rangle| \leq \sup \{ |\langle x, y \rangle| : x \in A \}$ for all $y \in F$. $A$ is Contained in its Bipolar if $A \subseteq E$ then $A \subseteq A^{\circ \circ}$. Classification of Bipolar of $A$ If $A \subseteq E$ then $A^{\circ \circ}$ is the $\sigma(G, F)$-closed absolutely convex hull of $A$, i.e, $\displaystyle{A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}^{\sigma(G, F)}}$. Classification of the Bipolar of $A$ (in $E$) If $E$ is a Hausdorff locally convex topological vector space (so that $(E, E')$ and $(E', E)$ are dual pairs replacing $(E, F)$ and $(F, G)$ above) and if $A \subseteq E$ then the bipolar $A^{\circ \circ}$ (in $E$) is the closed absolutely convex hull of $A$, i.e., $A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}$.
(4)
\begin{align} \quad \left ( \bigcap_{\alpha} A_{\alpha} \right )^{\circ} = \overline{\mathrm{abs \: conv}\left ( \bigcup_{\alpha} A_{\alpha}^{\circ} \right )}^{\sigma(F, E)} \end{align}

## 2.5. Review of Transposes

Let $(E, F)$ and $(G, H)$ be dual pairs.

• Recall from The Transpose of a Linear Operator page that if $t : E \to G$ is a linear operator, then the Transpose of $t$ is the linear operator $t' : H \to E^*$ defined by $h \mapsto t'(h)$ where for each $e \in E$ we have that $t'(h)(e) := \langle t(e), h \rangle$. We note that $t'$ has the special identity that for all $e \in E$ and for all $h \in H$:
(5)
\begin{align} \quad \langle e, t'(h) \rangle = \langle t(e), h \rangle \end{align}
• On the Weakly Continuous Linear Operators page we said that $t : E \to G$ is Weakly Continuous if it is continuous when $E$ is equipped with the $\sigma(E, F)$ topology and when $G$ is equipped with the $\sigma(G, H)$ topology.
• We then characterized weak continuity of linear operators:

If $(E, F)$ and $(G, H)$ are dual pairs and $t : E \to G$ is a linear operator then $t$ is weakly continuous if and only if $t'(H) \subseteq F$.

If $E$ and $F$ are Hausdorff locally convex topological vector spaces (so that $(E, E')$ and $(F, F')$ are dual pairs) and if $t : E \to F$ is a continuous linear operator then $t$ is weakly continuous.

• We remarked that the converse of the result above is not true in general.

If $(E, F)$ and $(G, H)$ are dual pairs, $A \subseteq E$, and $t : E \to G$ is weakly continuous, then:

(6)
\begin{align} \quad (t(A))^{\circ} = t'^{-1}(A^{\circ}) \end{align}

Where $(t(A))^{\circ}$ is the polar of $t(A)$ (in $H$), and $A^{\circ}$ is the polar of $A$ (in $F$).

## 2.6. Review of Finite-Dimensional Vector Spaces

(7)
\begin{align} \quad x := \lambda_1 e_1 + \lambda_2 e_2 + ... + \lambda_n e_n \end{align}
• The Dual Basis for $E^*$ is the basis $\{ e_1^*, e_2^*, ..., e_n^* \}$ where for each $1 \leq i \leq n$ we define:
(8)
\begin{align} \quad \langle x, e_i^* \rangle := \lambda_i \end{align}

If $E$ is a finite-dimensional vector space then there is a unique topology for which $E$ because a Hausdorff and locally convex topological vector space.

If $E$ is a Hausdorff locally convex topological vector space and if $M$ is a finite-dimensional subspace of $E$ then $M$ is closed.