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 $x \in E$:
2.2. Review of Linear Forms and the Hahn-Banach Theorem
- Recall from the Linear Forms on a Vector Space and its Algebraic Dual page that if $E$ is a vector space then a Linear Form (or Linear Functional) on $E$ is a linear operator from $E$ to the field $\mathbb{F}$ (of real or complex numbers).
- 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.
- On the Continuous Linear Forms on a TVS and its Continuous Dual and Closed Preimage Criterion for a Linear Form to be Continuous in a TVS pages we defined the Topological Dual (of a topological vector space $E$) to be $E'$, the set of all continuous linear forms on $E$. We summarize equivalent criterion for a linear form on a topological vector space $E$ to be continuous:
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 Closures of Subspaces of a Topological Vector Space page we proved that if $E$ is a topological vector space and if $M$ is a subspace of $E$ then the closure $\overline{M}$ is also a subspace of $E$.
- 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 for Vector Spaces Part 1
- The Hahn-Banach Theorem for Vector Spaces Part 2
- The Hahn-Banach Theorem for Vector Spaces Part 3
- The Hahn-Banach Theorem for Vector Spaces Part 4
- 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.
- On the Corollaries to the Hahn-Banach Theorem for Vector Spaces we looked at some corollaries to the Hahn-Banach theorem which are summarized below.
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 $a = o$. |
- Lastly, on the The Hahn-Banach Separation Theorem page we proved the important Hahn-Banach Separation Theorem:
(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:
- 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 $y_1, y_2, ..., y_n \in F$. |
- On the The Topological Dual of E Equipped with σ(E, F) is F page we classified the topological dual of $E$ equipped with the $\sigma(E, F)$ topology:
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:
- 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 $M^{\perp}$.
- On the The Polar Criterion for Equicontinuity of a Set of Linear Forms page, for Hausdorff locally convex topological vector spaces we classified the notion of equicontinuity in terms of polar sets:
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:
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair, A ⊆ A°°
- If E ⊆ G ⊆ F* and (E, F) is a Dual Pair then A°° is the σ(G, F)-Closed Absolutely Convex Hull of A
- If E is a Hausdorff LCTVS then A°° (in E) is the Closed Absolutely Convex Hull of A
- 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., $A^{\circ \circ} = \overline{\mathrm{abs \: conv}(A)}$. |
- Lastly, on The Polar of an Intersection of σ(E, F)-Closed Absolutely Convex Sets page we proved that if $\{ A_{\alpha} \}_{\alpha}$ is a collection of $\sigma(E, F)$-closed absolutely convex subsets of $E$ then:
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 $h \in H$:
- 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 $t'(H) \subseteq F$.
- On the For Hausdorff LCTVS E and F, if t from E to F is Continuous then t is Weakly Continuous page, we immediately obtained the following:
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.
- Lastly, on the For Dual Pairs (E, F), (G, H), and a Weakly Continuous Linear Operator t from E to G, (t(A))° = t'-1(A°) page we proved the following result for polars of images of weakly continuous linear operators.
If $(E, F)$ and $(G, H)$ are dual pairs, $A \subseteq E$, and $t : E \to G$ is weakly continuous, then:
(6)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
- Recall from The Dual Base for E* of a Finite-Dimensional Vector Space E page that if $E$ is a finite-dimensional vector space with $\mathrm{dim}(E) = n$ and if $\{ e_1, e_2, ..., e_n \}$ is a basis for $E$, then every $x \in E$ can be written uniquely in the form:
- The Dual Basis for $E^*$ is the basis $\{ e_1^*, e_2^*, ..., e_n^* \}$ where for each $1 \leq i \leq n$ we define:
- On the Finite-Dimensional Vector Spaces Have a Unique Locally Convex and Hausdorff Topology page we proved the following important result:
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.
- Lastly, on the Finite-Dimensional Subspaces are Closed in a Hausdorff LCTVS page we proved that:
If $E$ is a Hausdorff locally convex topological vector space and if $M$ is a finite-dimensional subspace of $E$ then $M$ is closed.