From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- .../alaoglu-bourbaki-theorem.md | 28 ++++++++++++++++++++++ 1 file changed, 28 insertions(+) create mode 100644 pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md (limited to 'pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md') diff --git a/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md new file mode 100644 index 0000000..be424c3 --- /dev/null +++ b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md @@ -0,0 +1,28 @@ +--- +title: Alaoglu–Bourbaki Theorem +parent: Locally Convex Spaces +grand_parent: More Functional Analysis +nav_order: 1 +--- + +# {{ page.title }} + +Let $X$ be locally convex space and +let $U \subset X$ be a neighborhood of zero. +Let $X'$ denote the continuous dual of $X$. +Recall that there is a canonical pairing + +$$ +X \times X' \to \CC, \quad (x,f) \mapsto \angles{x,f} = f(x). +$$ + +The weak topology on $X'$ with respect to this pairing +is called weak\* topology. +It is the weakest topology on $X'$ such that +all evaluation maps $\angles{x,\cdot} : X \to \CC$ are continuous. +The polar of $U$ is the subset $U^{\circ} \subset X'$. +The theorem asserts that $U^{\circ}$ is compact in the weak\* topology. + +{% theorem * Alaoglu–Bourbaki Theorem %} +The polar of a neighborhood of zero in a locally convex space is weak\* compact. +{% endtheorem %} -- cgit v1.2.3-54-g00ecf