Mixing
Hahn Banach Theorem, First geometric form
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I am studying the proof of Hahn Banach Theorem (First geometric form). And the following is a part of the proof.
Let $Usubseteq E$ be open, convex and nonempty and let $x_0in Ebackslash U$. Then, there exists $Fin E^{*}$ such that $F(x)<F(x_0)$ for all $xin U.$
Proof (in part)
Consider the subspace generated by $x_0in Ebackslash U$, which is given by
begin{align} M={x:,x=lambda x_0,,lambdain Bbb{R} } end{align}
Define begin{align} f:&Mto Bbb{R}\&xmapsto f(x)equiv f(lambda x_0)=lambda end{align}
Let $gamma, etain Bbb{R} $ and $x,yin E$, then there exists $lambda,betain Bbb{R}$ such that $x=lambda x_0$ and $y=beta x_0$ and
begin{align} f(gamma x+eta y)&=fleft(gamma (lambda x_0)+eta(beta x_0) right) \&=fleft((gamma lambda+etabeta)x_0 right)\&=gamma lambda+etabeta \&= gamma f( x)+eta f( y).end{align}
This implies that $f$ is linear on $M.$ Introducing the gauge $p$ on $U,$ we have that begin{align} f( x)leq p(x),;forall;xin M.end{align}
Then, we can apply the Hanh-Banach Theorem to find a linear functional $f:Eto Bbb{R},$ extending $f$ such that
begin{align} f( x)leq p(x),;forall;xin E.end{align}
The following is definition of gauge that I know:
Definition: Given that $E$ is a normed linear space. Let $Msubseteq E$ be an open, convex set with $0in M.$ For all $xin E$, define
begin{align} p(x)=inf{alpha>0:,alpha^{-1}xin M } end{align}
Then, $p$ is called a gauge of $M$.
Question: What kind of gauge $p,$ was defined on $U$ such that
begin{align} f( x)leq p(x),;forall;xin M?end{align}
functional-analysis analysis hahn-banach-theorem
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
$endgroup$
add a comment |
$begingroup$
I am studying the proof of Hahn Banach Theorem (First geometric form). And the following is a part of the proof.
Let $Usubseteq E$ be open, convex and nonempty and let $x_0in Ebackslash U$. Then, there exists $Fin E^{*}$ such that $F(x)<F(x_0)$ for all $xin U.$
Proof (in part)
Consider the subspace generated by $x_0in Ebackslash U$, which is given by
begin{align} M={x:,x=lambda x_0,,lambdain Bbb{R} } end{align}
Define begin{align} f:&Mto Bbb{R}\&xmapsto f(x)equiv f(lambda x_0)=lambda end{align}
Let $gamma, etain Bbb{R} $ and $x,yin E$, then there exists $lambda,betain Bbb{R}$ such that $x=lambda x_0$ and $y=beta x_0$ and
begin{align} f(gamma x+eta y)&=fleft(gamma (lambda x_0)+eta(beta x_0) right) \&=fleft((gamma lambda+etabeta)x_0 right)\&=gamma lambda+etabeta \&= gamma f( x)+eta f( y).end{align}
This implies that $f$ is linear on $M.$ Introducing the gauge $p$ on $U,$ we have that begin{align} f( x)leq p(x),;forall;xin M.end{align}
Then, we can apply the Hanh-Banach Theorem to find a linear functional $f:Eto Bbb{R},$ extending $f$ such that
begin{align} f( x)leq p(x),;forall;xin E.end{align}
The following is definition of gauge that I know:
Definition: Given that $E$ is a normed linear space. Let $Msubseteq E$ be an open, convex set with $0in M.$ For all $xin E$, define
begin{align} p(x)=inf{alpha>0:,alpha^{-1}xin M } end{align}
Then, $p$ is called a gauge of $M$.
Question: What kind of gauge $p,$ was defined on $U$ such that
begin{align} f( x)leq p(x),;forall;xin M?end{align}
functional-analysis analysis hahn-banach-theorem
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
$endgroup$
add a comment |
$begingroup$
I am studying the proof of Hahn Banach Theorem (First geometric form). And the following is a part of the proof.
Let $Usubseteq E$ be open, convex and nonempty and let $x_0in Ebackslash U$. Then, there exists $Fin E^{*}$ such that $F(x)<F(x_0)$ for all $xin U.$
Proof (in part)
Consider the subspace generated by $x_0in Ebackslash U$, which is given by
begin{align} M={x:,x=lambda x_0,,lambdain Bbb{R} } end{align}
Define begin{align} f:&Mto Bbb{R}\&xmapsto f(x)equiv f(lambda x_0)=lambda end{align}
Let $gamma, etain Bbb{R} $ and $x,yin E$, then there exists $lambda,betain Bbb{R}$ such that $x=lambda x_0$ and $y=beta x_0$ and
begin{align} f(gamma x+eta y)&=fleft(gamma (lambda x_0)+eta(beta x_0) right) \&=fleft((gamma lambda+etabeta)x_0 right)\&=gamma lambda+etabeta \&= gamma f( x)+eta f( y).end{align}
This implies that $f$ is linear on $M.$ Introducing the gauge $p$ on $U,$ we have that begin{align} f( x)leq p(x),;forall;xin M.end{align}
Then, we can apply the Hanh-Banach Theorem to find a linear functional $f:Eto Bbb{R},$ extending $f$ such that
begin{align} f( x)leq p(x),;forall;xin E.end{align}
The following is definition of gauge that I know:
Definition: Given that $E$ is a normed linear space. Let $Msubseteq E$ be an open, convex set with $0in M.$ For all $xin E$, define
begin{align} p(x)=inf{alpha>0:,alpha^{-1}xin M } end{align}
Then, $p$ is called a gauge of $M$.
Question: What kind of gauge $p,$ was defined on $U$ such that
begin{align} f( x)leq p(x),;forall;xin M?end{align}
functional-analysis analysis hahn-banach-theorem
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
$endgroup$
I am studying the proof of Hahn Banach Theorem (First geometric form). And the following is a part of the proof.
Let $Usubseteq E$ be open, convex and nonempty and let $x_0in Ebackslash U$. Then, there exists $Fin E^{*}$ such that $F(x)<F(x_0)$ for all $xin U.$
Proof (in part)
Consider the subspace generated by $x_0in Ebackslash U$, which is given by
begin{align} M={x:,x=lambda x_0,,lambdain Bbb{R} } end{align}
Define begin{align} f:&Mto Bbb{R}\&xmapsto f(x)equiv f(lambda x_0)=lambda end{align}
Let $gamma, etain Bbb{R} $ and $x,yin E$, then there exists $lambda,betain Bbb{R}$ such that $x=lambda x_0$ and $y=beta x_0$ and
begin{align} f(gamma x+eta y)&=fleft(gamma (lambda x_0)+eta(beta x_0) right) \&=fleft((gamma lambda+etabeta)x_0 right)\&=gamma lambda+etabeta \&= gamma f( x)+eta f( y).end{align}
This implies that $f$ is linear on $M.$ Introducing the gauge $p$ on $U,$ we have that begin{align} f( x)leq p(x),;forall;xin M.end{align}
Then, we can apply the Hanh-Banach Theorem to find a linear functional $f:Eto Bbb{R},$ extending $f$ such that
begin{align} f( x)leq p(x),;forall;xin E.end{align}
The following is definition of gauge that I know:
Definition: Given that $E$ is a normed linear space. Let $Msubseteq E$ be an open, convex set with $0in M.$ For all $xin E$, define
begin{align} p(x)=inf{alpha>0:,alpha^{-1}xin M } end{align}
Then, $p$ is called a gauge of $M$.
Question: What kind of gauge $p,$ was defined on $U$ such that
begin{align} f( x)leq p(x),;forall;xin M?end{align}
functional-analysis analysis hahn-banach-theorem
functional-analysis analysis hahn-banach-theorem
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
edited Jan 15 at 13:04
Omojola Micheal
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
asked Jan 15 at 12:22
Omojola MichealOmojola Micheal
1,877324
1,877324
add a comment |
add a comment |
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