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| In [[functional analysis]] and related areas of [[mathematics]] the '''weak topology''' is the [[coarser topology|coarsest]] [[polar topology]], the [[topology]] with the fewest [[open set]]s, on a [[dual pair]]. The [[finer topology|finest]] polar topology is called [[strong topology (polar topology)|strong topology]].
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| Under the weak topology the [[Bounded set (topological vector space)|bounded set]]s coincide with the [[relatively compact set]]s which leads to the important [[Bourbaki–Alaoglu theorem]].
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| ==Definition==
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| Given a [[dual pair]] <math>(X,Y,\langle , \rangle)</math> the '''weak topology''' <math>\sigma(X,Y)</math> is the weakest polar topology on <math>X</math> so that
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| : <math>(X,\sigma(X,Y))' \simeq Y</math>.
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| That is the [[continuous dual]] of <math>(X,\sigma(X,Y))</math> is equal to <math>Y</math> [[up to]] [[isomorphism]].
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| The weak topology is constructed as follows:
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| For every <math>y</math> in <math>Y</math> on <math>X</math> we define a [[semi norm]] on <math>X</math>
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| :<math>p_y:X \to \mathbb{R}</math> | |
| with
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| :<math>p_y(x) := \vert \langle x , y \rangle \vert \qquad x \in X </math>
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| This family of semi norms defines a [[locally convex topology]] on <math>X</math>.
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| ==Examples==
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| * Given a [[normed vector space]] <math>X</math> and its [[continuous dual]] <math>X'</math>, <math>\sigma(X, X')</math> is called the [[weak topology]] on <math>X</math> and <math>\sigma(X', X)</math> the [[weak-star topology|weak* topology]] on <math>X'</math>
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Weak Topology (Polar Topology)}}
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| [[Category:Topology of function spaces]]
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Oscar is how he's known as and he totally enjoys this title. Bookkeeping is what I do. South Dakota is where I've usually been residing. Doing ceramics is what my family members and I enjoy.
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