diff options
Diffstat (limited to 'pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md')
| -rw-r--r-- | pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md | 259 |
1 files changed, 155 insertions, 104 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 9d21d41..18cf64a 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 @@ -6,6 +6,7 @@ nav_order: 1 --- # {{ page.title }} +{: .no_toc } In fact, there are multiple theorems and corollaries which bear the name Hahn–Banach. @@ -13,111 +14,112 @@ All have in common that they guarantee the existence of linear functionals with various additional properties. -{: .definition-title } -> Definition (Sublinear Functional) -> -> A functional $p$ on a real vector space $X$ -> is called *sublinear* if it is -> {: .mb-0 } -> -> {: .mt-0 .mb-0 } -> - *positive-homogenous*, that is -> {: .mt-0 .mb-0 } -> -> $$ -> p(\alpha x) = \alpha \, p(x) \qquad \forall \alpha \ge 0, \ \forall x \in X, -> $$ -> -> - and satisfies the *triangle inequality* -> {: .mt-0 .mb-0 } -> -> $$ -> p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. -> $$ -> {: .katex-display .mb-0 } +<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 +{: .mb-0 } + +{: .mt-0 .mb-0 } +- *positively homogenous*, that is + {: .mt-0 .mb-0 } + + $$ + p(\alpha x) = \alpha \, p(x) \qquad \forall \alpha \ge 0, \ \forall x \in X, + $$ + +- and *subadditive*, that is + {: .mt-0 .mb-0 } + +$$ +p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. +$$ +{% enddefinition %} If $p$ is a sublinear functional, then $p(0)=0$ and $p(-x) \ge -p(x)$ for all $x$. Every norm on a real vector space is a sublinear functional. -{: .theorem-title } -> {{ page.title }} (Basic Version) -> -> Let $p$ be a sublinear functional on a real vector space $X$. -> Then there exists a linear functional $f$ on $X$ satisfying -> $f(x) \le p(x)$ for all $x \in X$. +{% theorem * Hahn–Banach Theorem (Basic Version) %} +Let $p$ be a sublinear functional on a real vector space $X$. +Then there exists a linear functional $f$ on $X$ satisfying +$f(x) \le p(x)$ for all $x \in X$. +{% endtheorem %} ## Extension Theorems -{: .theorem-title } -> {{ page.title }} (Extension, Real Vector Spaces) -> -> Let $p$ be a sublinear functional on a real vector space $X$. -> Let $f$ be a linear functional -> which is defined on a linear subspace $Z$ of $X$ -> and satisfies -> -> $$ -> f(x) \le p(x) \qquad \forall x \in Z. -> $$ -> -> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that -> -> $$ -> \tilde{f}(x) \le p(x) \qquad \forall x \in X. -> $$ +{% theorem * Hahn–Banach Theorem (Extension, Real Vector Spaces) %} +Let $p$ be a sublinear functional on a real vector space $X$. +Let $f$ be a linear functional +which is defined on a linear subspace $Z$ of $X$ +and satisfies + +$$ +f(x) \le p(x) \qquad \forall x \in Z. +$$ + +Then $f$ has a linear extension $\tilde{f}$ to $X$ such that + +$$ +\tilde{f}(x) \le p(x) \qquad \forall x \in X. +$$ +{% endtheorem %} {% proof %} {% endproof %} -{: .definition-title } -> Definition (Semi-Norm) -> -> We call a real-valued functional $p$ on a real or complex vector space $X$ -> a *semi-norm* if it is -> {: .mb-0 } -> -> {: .mt-0 .mb-0 } -> - *absolutely homogenous*, that is -> {: .mt-0 .mb-0 } -> -> $$ -> p(\alpha x) = \abs{\alpha} \, p(x) \qquad \forall \alpha \in \KK \ \forall x \in X, -> $$ -> - and satisfies the *triangle inequality* -> {: .mt-0 .mb-0 } -> -> $$ -> p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. -> $$ -> {: .katex-display .mb-0 } - -{: .theorem-title } -> {{ page.title }} (Extension, Real and Complex Vector Spaces) -> -> Let $p$ be a semi-norm on a real or complex vector space $X$. -> Let $f$ be a linear functional -> which is defined on a linear subspace $Z$ of $X$ -> and satisfies -> -> $$ -> \abs{f(x)} \le p(x) \qquad \forall x \in Z. -> $$ -> -> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that -> -> $$ -> \abs{\tilde{f}(x)} \le p(x) \qquad \forall x \in X. -> $$ - -{: .theorem-title } -> {{ page.title }} (Extension, Normed Spaces) -> -> Let $X$ be a real or complex normed space -> and let $f$ be a bounded linear functional -> defined on a linear subspace $Z$ of $X$. -> Then $f$ has a bounded linear extension $\tilde{f}$ to $X$ such that $\norm{\tilde{f}} = \norm{f}$. +{% definition Semi-Norm %} +We call a real-valued functional $p$ on a real or complex vector space $X$ +a *semi-norm* if it is +{: .mb-0 } + +{: .mt-0 .mb-0 } +- *absolutely homogenous*, that is + {: .mt-0 .mb-0 } + + $$ + p(\alpha x) = \abs{\alpha} \, p(x) \qquad \forall \alpha \in \KK \ \forall x \in X, + $$ +- and satisfies the *triangle inequality* + {: .mt-0 .mb-0 } + + $$ + p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. + $$ +{% enddefinition %} + +{% theorem * Hahn–Banach Theorem (Extension, Real and Complex Vector Spaces) %} +Let $p$ be a semi-norm on a real or complex vector space $X$. +Let $f$ be a linear functional +which is defined on a linear subspace $Z$ of $X$ +and satisfies + +$$ +\abs{f(x)} \le p(x) \qquad \forall x \in Z. +$$ + +Then $f$ has a linear extension $\tilde{f}$ to $X$ such that + +$$ +\abs{\tilde{f}(x)} \le p(x) \qquad \forall x \in X. +$$ +{% endtheorem %} + +{% theorem * Hahn–Banach Theorem (Extension, Normed Spaces) %} +Let $X$ be a real or complex normed space +and let $f$ be a bounded linear functional +defined on a linear subspace $Z$ of $X$. +Then $f$ has a bounded linear extension $\tilde{f}$ to $X$ such that $\norm{\tilde{f}} = \norm{f}$. +{% endtheorem %} {% proof %} We apply the preceding theorem with $p(x) = \norm{f} \norm{x}$ @@ -131,17 +133,66 @@ Corollaries Important consequence: canonical embedding into bidual +{% theorem * Hahn–Banach Theorem (Existence of Functionals) %} +Let $X$ be a real or complex normed space +and let $x$ be a nonzero element of $X$. +Then there exists a bounded linear functional $f$ on $X$ +with $f(x) = \norm{x}$ and $\norm{f} = 1$. +{% endtheorem %} + +{% proof %} +On the linear subspace $\KK x \subset X$ spanned by $x$ +we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \in \KK$. +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 $\norm{f} = \norm{f_0} = 1$. +{% endproof %} + +Recall that for a normed space $X$ we denote its (topological) dual space by $X'$. + +{% corollary %} +For every element $x$ of a real or complex normed space $X$ one has + +$$ +\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} +$$ + +and the supremum is attained. +{% endcorollary %} + +{% corollary %} +The elements of a real or complex normed space $X$ +are separated by the elements of its dual $X'$. +{% endcorollary %} + ## Separation Theorems -{: .theorem-title } -> {{ page.title }} (Separation, Point and Closed Subspace) -> -> Suppose $Z$ is a closed subspace -> of a normed space $X$ and $x$ lies in $X \setminus Z$. -> Then there exists a bounded linear functional on $X$ -> which vanishes on $Z$ but has a nonzero value at $x$. - -{: .theorem-title } -> {{ page.title }} (Separation, Convex Sets) -> -> TODO +{% theorem * Hahn–Banach Theorem (Separation, Point and Closed Subspace) %} +Suppose $Z$ is a closed subspace of a normed space $X$ +and $x$ lies in $X \setminus Z$. +Then there exists a bounded linear functional $f$ on $X$ +vanishing on $Z$ and with nonzero value $f(x) = \dist{x,Z}$. +{% endtheorem %} + +{% proof %} +Since $Z$ is a closed subspace of $X$, +the quotient vector space $X/Z$ becomes a normed space +with the quotient norm given by + +$$ +\norm{y + Z} = \dist{y,Z} = \inf_{z \in Z} \norm{y-z} \quad \forall y \in X. +$$ + +Moreover, the canonical mapping $\pi : X \to X/Z$, $y \mapsto y+Z$, is bounded. +Given a $x \in X$ that does not lie in $Z$, the null space of $\pi$, +we see that $\pi(x)$ is a nonzero element of $X/Z$. +By Hahn–Banach, there exists a bounded linear functional $g$ on $X/Z$ +with $g(\pi(x)) = \norm{x} = \dist{x,Z} \ne 0$. +Now the composition $f = g \circ \pi$ is a bounded functional on $X$ +with the desired properties. +{% endproof %} + +{% theorem * Hahn–Banach Theorem (Separation, Convex Sets) %} +TODO +{% endtheorem %} |