Editing Weak topology (section)

=== Weak topology with respect to a pairing ===
{{Main|Dual system#Weak topology}}

Both the weak topology and the weak* topology are special cases of a more general construction for [[Dual system|pairings]], which we now describe. 
The benefit of this more general construction is that any definition or result proved for it applies to ''both'' the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of this more general construction.

Suppose {{math|(''X'', ''Y'', ''b'')}} is a [[Dual system|pairing]] of vector spaces over a topological field <math>\mathbb{K}</math> (i.e. {{mvar|X}} and {{mvar|Y}} are vector spaces over <math>\mathbb{K}</math> and {{math|''b'' : ''X'' × ''Y'' → <math>\mathbb{K}</math>}} is a [[bilinear map]]).

:'''Notation.''' For all {{math|''x'' ∈ ''X''}}, let {{math|''b''(''x'', •) : ''Y'' → <math>\mathbb{K}</math>}} denote the linear functional on {{mvar|Y}} defined by {{math|''y'' {{mapsto}} ''b''(''x'', ''y'')}}. Similarly, for all {{math|''y'' ∈ ''Y''}}, let {{math|''b''(•, ''y'') : ''X'' → <math>\mathbb{K}</math>}} be defined by {{math|''x'' {{mapsto}} ''b''(''x'', ''y'')}}.

:'''Definition.''' The '''weak topology on {{mvar|X}}''' induced by {{mvar|Y}} (and {{mvar|b}}) is the weakest topology on {{mvar|X}}, denoted by {{math|𝜎(''X'', ''Y'', ''b'')}} or simply {{math|𝜎(''X'', ''Y'')}}, making all maps {{math|''b''(•, ''y'') : ''X'' → <math>\mathbb{K}</math>}} continuous, as {{mvar|y}} ranges over {{mvar|Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

The weak topology on {{mvar|Y}} is now automatically defined as described in the article [[Dual system]]. However, for clarity, we now repeat it.

:'''Definition.''' The '''weak topology on {{mvar|Y}}''' induced by {{mvar|X}} (and {{mvar|b}}) is the weakest topology on {{mvar|Y}}, denoted by {{math|𝜎(''Y'', ''X'', ''b'')}} or simply {{math|𝜎(''Y'', ''X'')}}, making all maps {{math|''b''(''x'', •) : ''Y'' → <math>\mathbb{K}</math>}} continuous, as {{mvar|x}} ranges over {{mvar|X}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

If the field <math>\mathbb{K}</math> has an [[absolute value]] {{math|{{mabs|⋅}}}}, then the weak topology {{math|𝜎(''X'', ''Y'', ''b'')}} on {{mvar|X}} is induced by the family of [[seminorm]]s, {{math|''p''<sub>''y''</sub> : ''X'' → <math>\mathbb{R}</math>}}, defined by

:{{math|''p''<sub>''y''</sub>(''x'') :{{=}} {{mabs|''b''(''x'', ''y'')}}}}

for all {{math|''y'' &isin; ''Y''}} and {{math|''x'' &isin; ''X''}}. This shows that weak topologies are [[locally convex space|locally convex]].

:'''Assumption.''' We will henceforth assume that <math>\mathbb{K}</math> is either the [[real number]]s <math>\mathbb{R}</math> or the [[complex number]]s <math>\mathbb{C}</math>.

==== 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}}

==== The weak and weak* topologies ====

Let {{mvar|X}} be a [[topological vector space]] (TVS) over <math>\mathbb{K}</math>, that is, {{mvar|X}} is a <math>\mathbb{K}</math> [[vector space]] equipped with a [[topological space|topology]] so that vector addition and [[scalar multiplication]] are continuous. We call the topology that {{mvar|X}} starts with the '''original''', '''starting''', or '''given topology''' (the reader is cautioned against using the terms "[[initial topology]]" and "[[strong topology]]" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on {{mvar|X}} using the topological or [[continuous dual space]] <math>X^*</math>, which consists of all [[linear functional]]s from {{mvar|X}} into the base field <math>\mathbb{K}</math> that are [[continuous function (topology)|continuous]] with respect to the given topology.

Recall that <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 X^*</math>, where in particular, <math>\langle \cdot,x'\rangle=x'(\cdot)= x'</math>.

:'''Definition.''' The '''weak topology on {{mvar|X}}''' is the weak topology on {{mvar|X}} with respect to the [[Dual system#Canonical duality on a vector space|canonical pairing]] <math>\langle X,X^*\rangle</math>. That is, it is the weakest topology on {{mvar|X}} making all maps <math>x' =\langle\cdot,x'\rangle:X\to\mathbb{K}</math> continuous, as <math>x'</math> ranges over <math>X^*</math>.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

:'''Definition''': The '''weak topology on <math>X^*</math>''' is the weak topology on <math>X^*</math> with respect to the [[Dual system#Canonical duality on a vector space|canonical pairing]] <math>\langle X,X^*\rangle</math>. That is, it is the weakest topology on <math>X^*</math> making all maps <math>\langle x,\cdot\rangle:X^*\to\mathbb{K}</math> continuous, as {{mvar|x}} ranges over {{mvar|X}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} This topology is also called the '''weak* topology'''.

We give alternative definitions below.