Editing Weak topology (section)

==== Canonical duality ====

We now consider the special case where {{mvar|Y}} is a vector subspace of the [[algebraic dual space]] of {{mvar|X}} (i.e. a vector space of linear functionals on {{mvar|X}}).

There is a pairing, denoted by <math>(X,Y,\langle\cdot, \cdot\rangle)</math> or <math>(X,Y)</math>, called the [[Dual system#Canonical duality on a vector space|canonical pairing]] whose bilinear map <math>\langle\cdot, \cdot\rangle</math> is the '''canonical evaluation map''', defined by <math>\langle x,x'\rangle =x'(x)</math> for all <math>x\in X</math> and <math>x'\in Y</math>. Note in particular that <math>\langle \cdot,x'\rangle</math> is just another way of denoting <math>x'</math> i.e. <math>\langle \cdot,x'\rangle=x'(\cdot)</math>.

:'''Assumption.''' If {{mvar|Y}} is a vector subspace of the [[algebraic dual space]] of {{mvar|X}} then we will assume that they are associated with the canonical pairing {{math|{{angbr|''X'', ''Y''}}}}.

In this case, the '''weak topology on {{mvar|X}}''' (resp. the '''weak topology on {{var|Y}}'''), denoted by {{math|𝜎(''X'',''Y'')}} (resp. by {{math|𝜎(''Y'',''X'')}}) is the [[Dual system#Weak topology|weak topology]] on {{mvar|X}} (resp. on {{mvar|Y}}) with respect to the canonical pairing {{math|{{angbr|''X'', ''Y''}}}}.

The topology {{math|σ(''X'',''Y'')}} is the [[initial topology]] of {{mvar|X}} with respect to {{mvar|Y}}.

If {{mvar|Y}} is a vector space of linear functionals on {{mvar|X}}, then the continuous dual of {{mvar|X}} with respect to the topology {{math|σ(''X'',''Y'')}} is precisely equal to {{mvar|Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}{{harv|Rudin|1991|loc=Theorem 3.10}}