Editing Weak topology (section)

{{Short description|Mathematical term}}
{{about|the weak topology on a normed vector space|the weak topology induced by a general family of maps|initial topology|the weak topology generated by a cover of a space|coherent topology}}

In [[mathematics]], '''weak topology''' is an alternative term for certain [[initial topology|initial topologies]], often on [[topological vector space]]s or spaces of [[linear operator]]s, for instance on a [[Hilbert space]]. The term is most commonly used for the initial topology of a topological vector space (such as a [[normed vector space]]) with respect to its [[continuous dual space|continuous dual]]. The remainder of this article will deal with this case, which is one of the concepts of [[functional analysis]].

One may call subsets of a topological vector space '''weakly closed''' (respectively, '''weakly compact''', etc.) if they are [[closed set|closed]] (respectively, [[compact set|compact]], etc.) with respect to the weak topology. Likewise, functions are sometimes called '''[[Dual system#Weak continuity|weakly continuous]]''' (respectively, '''weakly differentiable''', '''weakly analytic''', etc.) if they are [[continuous function|continuous]] (respectively, [[derivative|differentiable]], [[analytic function|analytic]], etc.) with respect to the weak topology.