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Diffstat (limited to 'pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md')
| -rw-r--r-- | pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md | 4 |
1 files changed, 2 insertions, 2 deletions
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 |