Editing Weak topology (section)

==== 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.