Editing Dual space (section)

=== Topologies on the dual ===
{{Main|Polar topology|Dual system}}

There is a standard construction for introducing a topology on the continuous dual <math>V'</math> of a topological vector space <math>V</math>. Fix a collection <math>\mathcal{A}</math> of [[Bounded set (topological vector space)|bounded subsets]] of <math>V</math>.
This gives the topology on <math>V</math> of uniform convergence on sets from <math>\mathcal{A},</math> or what is the same thing, the topology generated by [[seminorm]]s of the form

:<math>\|\varphi\|_A = \sup_{x\in A} |\varphi(x)|,</math>

where <math>\varphi</math> is a continuous linear functional on <math>V</math>, and <math>A</math> runs over the class <math>\mathcal{A}.</math>

This means that a net of functionals <math>\varphi_i</math> tends to a functional <math>\varphi</math> in <math>V'</math> if and only if

:<math>\text{ for all } A\in\mathcal{A}\qquad \|\varphi_i-\varphi\|_A = \sup_{x\in A} |\varphi_i(x)-\varphi(x)|\underset{i\to\infty}{\longrightarrow} 0. </math>

Usually (but not necessarily) the class <math>\mathcal{A}</math> is supposed to satisfy the following conditions:
* Each point <math>x</math> of <math>V</math> belongs to some set <math>A\in\mathcal{A}</math>:
*:<math>\text{ for all } x \in V\quad \text{ there exists some } A \in \mathcal{A}\quad \text{ such that } x \in A.</math>
* Each two sets <math>A \in \mathcal{A}</math> and <math>B \in \mathcal{A}</math> are contained in some set <math>C \in \mathcal{A}</math>:
*:<math>\text{ for all } A, B \in \mathcal{A}\quad \text{ there exists some } C \in \mathcal{A}\quad \text{ such that } A \cup B \subseteq C.</math>
* <math>\mathcal{A}</math> is closed under the operation of multiplication by scalars: 
*:<math>\text{ for all } A \in \mathcal{A}\quad \text{ and all } \lambda \in {\mathbb F}\quad \text{ such that } \lambda \cdot A \in \mathcal{A}.</math>

If these requirements are fulfilled then the corresponding topology on <math>V'</math> is Hausdorff and the sets

:<math>U_A ~=~ \left \{ \varphi \in V' ~:~ \quad \|\varphi\|_A < 1 \right \},\qquad \text{ for } A \in \mathcal{A}</math>

form its local base.

Here are the three most important special cases.
* The [[Strong topology (polar topology)|strong topology]] on <math>V'</math> is the topology of uniform convergence on [[Bounded set (topological vector space)|bounded subsets]] in <math>V</math> (so here <math>\mathcal{A}</math> can be chosen as the class of all bounded subsets in <math>V</math>).
If <math>V</math> is a [[normed vector space]] (for example, a [[Banach space]] or a [[Hilbert space]]) then the strong topology on <math>V'</math> is normed (in fact a Banach space if the field of scalars is complete), with the norm
::<math>\|\varphi\| = \sup_{\|x\| \le 1 } |\varphi(x)|.</math>
* The [[stereotype space|stereotype topology]] on <math>V'</math> is the topology of uniform convergence on [[Totally bounded space|totally bounded sets]] in <math>V</math> (so here <math>\mathcal{A}</math>  can be chosen as the class of all totally bounded subsets in <math>V</math>).
* The [[weak topology]] on <math>V'</math> is the topology of uniform convergence on finite subsets in <math>V</math> (so here <math>\mathcal{A}</math>  can be chosen as the class of all finite subsets in <math>V</math>).

Each of these three choices of topology on <math>V'</math> leads to a variant of [[Reflexive space|reflexivity property]] for topological vector spaces:
* If <math>V'</math> is endowed with the [[strong topology]], then the corresponding notion of reflexivity is the standard one: the spaces reflexive in this sense are just called ''reflexive''.<ref name="H.Schaefer 1966 loc=IV.5.5">{{harvnb|Schaefer|1966|loc=IV.5.5}}</ref>
* If <math>V'</math> is endowed with the stereotype dual topology, then the corresponding reflexivity is presented in the theory of [[stereotype space]]s: the spaces reflexive in this sense are called ''stereotype''.
* If <math>V'</math> is endowed with the [[weak topology]], then the corresponding reflexivity is presented in the theory of [[dual pair]]s:<ref name="H.Schaefer 1966 loc=IV.1">{{harvnb|Schaefer|1966|loc=IV.1}}</ref> the spaces reflexive in this sense are arbitrary (Hausdorff) locally convex spaces with the weak topology.<ref name="H.Schaefer 1966 loc=IV.1.2">{{harvnb|Schaefer|1966|loc=IV.1.2}}</ref>