From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 15 Feb 2024 05:11:07 +0100 Subject: Update --- .../the-fundamental-four/open-mapping-theorem.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md') diff --git a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md index b191bb2..e7f2b70 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md @@ -30,10 +30,10 @@ This remains true, if we take closures: $\bigcup \overline{mTB_X} = Y$. Hence, we have written the space $Y$, which is assumed to have a complete norm, -as the union of countably many closed sets. It follows form the +as the union of countably many closed sets. It follows from the [Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %}) that $\overline{mTB_X}$ has nonempty interior for some $m$. -Thus there are $q \in Y$ and $\alpha > 0$ +Thus, there are $q \in Y$ and $\alpha > 0$ such that $q + \alpha B_Y \subset \overline{mTB_X}$. Choose a $p \in X$ with $Tp=q$. It is a well known fact, that in a normed space -- cgit v1.2.3-54-g00ecf