Duality and Weak Topologies

Definition 1.3.1   Let $X$ and $Y$ be vector spaces. We say $X$ and $Y$ are an algebraic dual pair (or, an algebraic duality) if there is a bilinear functional

$$\left\langle{ \cdot, \cdot }\right\rangle : X \times Y \rightarrow \mathbb{F}$$

so that

1. The collection of functionals
   $$\left\langle{ \cdot, y }\right\rangle : x \mapsto \left\langle{ x,y }\right\rangle , y \in Y$$
   separates points in $X$.
2. The collection of functionals
   $$\left\langle{ x, \cdot }\right\rangle : y \mapsto \left\langle{ x,y }\right\rangle , x \in X$$
   separates points in $Y$.

Example:
$X$ and $X^*$ are dual pair via $(x,f) \mapsto f(x)$.
If $X, Y$ are LCS and all the functionals are continuous , we drop the adjective ``algebraic"

Wednesday, January 17, 2006:

More on duality and the weak topologies.

Recall: For a vector space $X$ and a collection of linear functionals $F$, on $X$, the weak topology induced by $F$ is the LCS topology induced by the seminorms

$$\rho_f(x) = \vert f(x)\vert, f \in F$$

This is denoted $\sigma(X,F)$ or $\sigma(X,F)$-topology.

Definition 1.3.2   If $X$ is a LCS, the weak topology on $X$ is $\sigma(X,X^*)$ (also denoted $wk$ in the text) and the weak star topology on $X^*$ is $\sigma(X^*,X)$ (where $X$ is thought of as a subset of $X^{**}$). The book denotes this topology as $wk^*$.

``Obvious" Properties

1. If $F \subset G$, sets of functionals, then $\sigma(X,F) \subset \sigma(X,G)$.
2. $\sigma(X,F)$ is Hausdorff iff $F$ separates points; i.e.,
   $$\bigcap_{f \in F} \ker f = \{ 0 \}$$

3. If $f \in F$, then $f$ is continuous with respect to the $\sigma(X,F)$ topology.
   Proof.

   $$f^{-1}(B_\epsilon (0)) = \vert f\vert^{-1}((-\epsilon ,\epsilon )) = \{ x \in X : p_f(x) < \epsilon \}$$

   The set on the right is open by the definition of weak topology. $\qedsymbol$

4. Suppose $V$ is a neighborhood of 0 in the $\sigma(X,F)$-topology. If $x$ is infinite-dimensional, then $V$ contains an infinite-dimensional subspace of $X$.
   Proof. $V$ contains a basic open set containing 0. Thus, there is $f_1, \ldots, f_n \in F$ and $\epsilon _1, \ldots, \epsilon _n > 0$ such that

   $$V \supset \{ x in X : \vert f_i(x)\vert < \epsilon _i, i = 1, \ldots, n \}$$

   If $N = \{ x \in X : f_i(x) = 0, i = 1, \ldots, n \}$ then $N \leq V$ and $\dim X \leq n + \dim N$. $\qedsymbol$

Fact: For $\phi, \phi_1, \ldots, \phi_n$ linear functionals on a vector space $X$, the following are equivalent:

1. $$\phi = \sum_{i=1}^n \alpha_i \phi_i$$ for scalars $\alpha_1, \ldots, \alpha_n$
2. There is $\alpha \geq 0$ such that $\vert\phi(x)\vert \leq \alpha \max \vert \phi_i(x)\vert$
3. $\bigcap \ker \phi_i \subset \ker \phi$.

Proof. We prove two different parts:

$1 \implies 2 \implies 3$

Are either clear or repeated computations.

$3 \implies 1$

Define $T:X \rightarrow \mathbb{F}^n$ by

$$T(x) = (\phi_1 x, \phi_2 x, \ldots, \phi_n x).$$

Observe $\ker T = \bigcap \ker \phi_i \subset \ker \phi$.
By linear algebra, there is $f: \mathbb{F}^n \rightarrow \mathbb{F}$, a linear functional such that $\phi(x) = f \circ T(x)$. But $f$ is given by a matrix $[ \alpha_1, \ldots, \alpha_n]$ such that $f(\beta_1, \ldots, \beta_n) = \sum_{i=1}^n \alpha_i \beta_i$
Thus,

$$\phi(x) = f \circ T(x) = f(\phi_1 x_1, \ldots, \phi_n x)$$

$$= \sum_{i=1}^n \alpha_i \phi_i(x)$$

So, (1) holds.

$\qedsymbol$

Theorem 1.3.3   Fundamental Theorem of Weak Topologies:
If a linear functional $\phi$ on $X$ is continuous in the $\sigma(X,F)$ topology, then $\phi$ is in the linear manifold generated by $F$.

Proof. By the first result for dual spaces, there are seminorms $p_{f_1}, \ldots, p_{f_n}$, $f_i \in F$ and $\alpha_1, \ldots, \alpha_n \geq 0$ so that

$$\vert\phi(x)\vert \leq \sum_{i=1}^n \alpha_i p_{f_i}(x)$$

Letting $\alpha = \sum \alpha_i$, we have

$$\vert\phi(x)\vert \leq \alpha \max \vert f_i(x)\vert$$

By the fact just proved, $\phi$ is a linear combination of the $f_i$. $\qedsymbol$

Corollary 1.3.4   These corollaries follow from the above theorem:

1. (Universal property of the weak topology) If $F$ is a linear space of functionals on $X$ then, with the $\sigma(X,F)$ topology, $X^*$ is $F$. In fact, $\sigma(X,F)$ is the smallest topology with this property. (Easy to prove from the definition)
   In particular, for $X$ an LCS,
   $$(X,wk)^* = X^*,$$
   $$(X^*, wk^*)^* = X.$$

Concretely, thinking of $\ell^1$ as $(c_0)^*$,

$$(\ell^1,wk^*)^* = c_0$$

but $(c_0)^{**} = \ell^\infty$

Assignment In #6, complete means (i.e., $(x_n)$ Cauchy), for all $U$ open, $0 \in U$, there exists $N$ such that for all $m,n \geq N$,

$$x_m - x_n \in U$$

Friday, January 20, 2006:

More corollaries of the fundamental theorem:

1. If $L$ is the linear manifold generated by $F$, then the $\sigma(X,F)$ and $\sigma(X,L)$ topologies are the same.
2. If $X, Y$ are vector spaces that are an algebraic dual pair, i.e., we have $\left\langle{ \cdot, \cdot }\right\rangle : X \times Y \rightarrow \mathbb{F}$ and we give $X$ and $Y$ the weak topologies induced by $\left\langle{ \cdot, Y }\right\rangle$ and $\left\langle{ X, \cdot }\right\rangle$, then $X^* = Y$ and $Y^* = X$.