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Diffstat (limited to 'pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md')
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1 files changed, 2 insertions, 12 deletions
diff --git a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md index 18cf64a..a2602ac 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md @@ -6,7 +6,6 @@ nav_order: 1 --- # {{ page.title }} -{: .no_toc } In fact, there are multiple theorems and corollaries which bear the name Hahn–Banach. @@ -14,15 +13,6 @@ All have in common that they guarantee the existence of linear functionals with various additional properties. -<details open markdown="block"> - <summary> - Table of contents - </summary> - {: .text-delta } -- TOC -{:toc} -</details> - {% definition Sublinear Functional %} A functional $p$ on a real vector space $X$ is called *sublinear* if it is @@ -146,7 +136,7 @@ we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \i It is easy to check that $f_0$ is linear and bounded with norm $\norm{f_0} = 1$. By the Hahn–Banach Extension Theorem for Normed Spaces, there exists a bounded linear functional $f$ on $X$ extending $f_0$ with identical norm. -Hence we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$. +Hence, we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$. {% endproof %} Recall that for a normed space $X$ we denote its (topological) dual space by $X'$. @@ -155,7 +145,7 @@ Recall that for a normed space $X$ we denote its (topological) dual space by $X' For every element $x$ of a real or complex normed space $X$ one has $$ -\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} +\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} $$ and the supremum is attained. |