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| In [[functional analysis]] and related areas of [[mathematics]] the '''strong topology''' is the [[finer topology|finest]] [[polar topology]], the [[topology]] with the most [[open set]]s, on a [[dual pair]]. The [[coarser topology|coarsest]] polar topology is called [[weak topology (polar topology)|weak topology]].
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| == Definition ==
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| Let <math>(X,Y,\langle , \rangle)</math> be a [[dual pair]] of vector spaces over the field <math>{\mathbb F}</math> of real (<math>{\mathbb R}</math>) or complex (<math>{\mathbb C}</math>) numbers. Let us denote by <math>{\mathcal B}</math> the system of all subsets <math>B\subseteq X</math> bounded by elements of <math>Y</math> in the following sense:
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| :<math>
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| \forall y\in Y \qquad \sup_{x\in B}|\langle x, y\rangle|<\infty.
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| </math>
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| Then the '''strong topology''' <math>\beta(Y,X)</math> on <math>Y</math> is defined as the locally convex topology on <math>Y</math> generated by the seminorms of the form
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| :<math>
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| ||y||_B=\sup_{x\in B}|\langle x, y\rangle|,\qquad y\in Y,\qquad B\in{\mathcal B}.
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| </math>
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| In the special case when <math>X</math> is a [[locally convex space]], the '''strong topology''' on the (continuous) [[continuous dual space|dual space]] <math>X'</math> (i.e. on the space of all continuous linear functionals <math>f:X\to{\mathbb F}</math>) is defined as the strong topology <math>\beta(X',X)</math>, and it coincides with the topology of uniform convergence on [[bounded set]]s in <math>X</math>, i.e. with the topology on <math>X'</math> generated by the seminorms of the form
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| :<math>
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| ||f||_B=\sup_{x\in B}|f(x)|,\qquad f\in X',
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| </math>
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| where <math>B</math> runs over the family of all [[bounded set]]s in <math>X</math>. The space <math>X'</math> with this topology is called '''strong dual space''' of the space <math>X</math> and is denoted by <math>X'_\beta</math>.
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| == Examples ==
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| * If <math>X</math> is a [[normed vector space]], then its (continuous) [[continuous dual|dual space]] <math>X'</math> with the strong topology coincides with the [[Banach space#dual space|Banach dual space]] <math>X'</math>, i.e. with the space <math>X'</math> with the topology induced by the [[operator norm]]. Conversely <math>\beta(X, X')</math>-topology on <math>X</math> is identical to the topology induced by the [[norm (mathematics)|norm]] on <math>X</math>.
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| == Properties ==
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| * If <math>X</math> is a [[barrelled space]], then its topology coincides with the strong topology <math>\beta(X,X')</math> on <math>X</math> and with the [[Mackey topology]] on <math>X</math> generated by the pairing <math>(X,X')</math>.
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| == References ==
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| * {{cite book
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| | last = Schaefer
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| | first = Helmuth H.
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| <!-- | authorlink = Helmuth Schaefer -->
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| | year = 1966
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| | title = Topological vector spaces
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| | series=
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| | volume=
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| | publisher = The MacMillan Company
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| | location = New York
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| | isbn = 0-387-98726-6
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| }}
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| [[Category:Topology of function spaces]]
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| {{Functional Analysis}}
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| {{mathanalysis-stub}}
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