(* Title: HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>The supremum wrt.\ the function order\<close>
theory Hahn_Banach_Sup_Lemmas
imports Function_Norm Zorn_Lemma
begin
text \<open>
This section contains some lemmas that will be used in the proof of the
Hahn-Banach Theorem. In this section the following context is presumed. Let
\<open>E\<close> be a real vector space with a seminorm \<open>p\<close> on \<open>E\<close>. \<open>F\<close> is a subspace of
\<open>E\<close> and \<open>f\<close> a linear form on \<open>F\<close>. We consider a chain \<open>c\<close> of norm-preserving
extensions of \<open>f\<close>, such that \<open>\<Union>c = graph H h\<close>. We will show some properties
about the limit function \<open>h\<close>, i.e.\ the supremum of the chain \<open>c\<close>.
\<^medskip>
Let \<open>c\<close> be a chain of norm-preserving extensions of the function \<open>f\<close> and let
\<open>graph H h\<close> be the supremum of \<open>c\<close>. Every element in \<open>H\<close> is member of one of
the elements of the chain.
\<close>
lemmas [dest?] = chainsD
lemmas chainsE2 [elim?] = chainsD2 [elim_format]
lemma some_H'h't:
assumes M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and u: "graph H h = \c"
and x: "x \ H"
shows "\H' h'. graph H' h' \ c
\<and> (x, h x) \<in> graph H' h'
\<and> linearform H' h' \<and> H' \<unlhd> E
\<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h'
\<and> (\<forall>x \<in> H'. h' x \<le> p x)"
proof -
from x have "(x, h x) \ graph H h" ..
also from u have "\ = \c" .
finally obtain g where gc: "g \ c" and gh: "(x, h x) \ g" by blast
from cM have "c \ M" ..
with gc have "g \ M" ..
also from M have "\ = norm_pres_extensions E p F f" .
finally obtain H' and h' where g: "g = graph H' h'"
and * : "linearform H' h'" "H' \ E" "F \ H'"
"graph F f \ graph H' h'" "\x \ H'. h' x \ p x" ..
from gc and g have "graph H' h' \ c" by (simp only:)
moreover from gh and g have "(x, h x) \ graph H' h'" by (simp only:)
ultimately show ?thesis using * by blast
qed
text \<open>
\<^medskip>
Let \<open>c\<close> be a chain of norm-preserving extensions of the function \<open>f\<close> and let
\<open>graph H h\<close> be the supremum of \<open>c\<close>. Every element in the domain \<open>H\<close> of the
supremum function is member of the domain \<open>H'\<close> of some function \<open>h'\<close>, such
that \<open>h\<close> extends \<open>h'\<close>.
\<close>
lemma some_H'h':
assumes M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and u: "graph H h = \c"
and x: "x \ H"
shows "\H' h'. x \ H' \ graph H' h' \ graph H h
\<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
\<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
proof -
from M cM u x obtain H' h' where
x_hx: "(x, h x) \ graph H' h'"
and c: "graph H' h' \ c"
and * : "linearform H' h'" "H' \ E" "F \ H'"
"graph F f \ graph H' h'" "\x \ H'. h' x \ p x"
by (rule some_H'h't [elim_format]) blast
from x_hx have "x \ H'" ..
moreover from cM u c have "graph H' h' \ graph H h" by blast
ultimately show ?thesis using * by blast
qed
text \<open>
\<^medskip>
Any two elements \<open>x\<close> and \<open>y\<close> in the domain \<open>H\<close> of the supremum function \<open>h\<close>
are both in the domain \<open>H'\<close> of some function \<open>h'\<close>, such that \<open>h\<close> extends
\<open>h'\<close>.
\<close>
lemma some_H'h'2:
assumes M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and u: "graph H h = \c"
and x: "x \ H"
and y: "y \ H"
shows "\H' h'. x \ H' \ y \ H'
\<and> graph H' h' \<subseteq> graph H h
\<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
\<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
proof -
txt \<open>\<open>y\<close> is in the domain \<open>H''\<close> of some function \<open>h''\<close>, such that \<open>h\<close>
extends \<open>h''\<close>.\<close>
from M cM u and y obtain H' h' where
y_hy: "(y, h y) \ graph H' h'"
and c': "graph H' h' \ c"
and * :
"linearform H' h'" "H' \ E" "F \ H'"
"graph F f \ graph H' h'" "\x \ H'. h' x \ p x"
by (rule some_H'h't [elim_format]) blast
txt \<open>\<open>x\<close> is in the domain \<open>H'\<close> of some function \<open>h'\<close>,
such that \<open>h\<close> extends \<open>h'\<close>.\<close>
from M cM u and x obtain H'' h'' where
x_hx: "(x, h x) \ graph H'' h''"
and c'': "graph H'' h'' \ c"
and ** :
"linearform H'' h''" "H'' \ E" "F \ H''"
"graph F f \ graph H'' h''" "\x \ H''. h'' x \ p x"
by (rule some_H'h't [elim_format]) blast
txt \<open>Since both \<open>h'\<close> and \<open>h''\<close> are elements of the chain, \<open>h''\<close> is an
extension of \<open>h'\<close> or vice versa. Thus both \<open>x\<close> and \<open>y\<close> are contained in
the greater one. \label{cases1}\<close>
from cM c'' c' consider "graph H'' h'' \ graph H' h'" | "graph H' h' \ graph H'' h''"
by (blast dest: chainsD)
then show ?thesis
proof cases
case 1
have "(x, h x) \ graph H'' h''" by fact
also have "\ \ graph H' h'" by fact
finally have xh:"(x, h x) \ graph H' h'" .
then have "x \ H'" ..
moreover from y_hy have "y \ H'" ..
moreover from cM u and c' have "graph H' h' \ graph H h" by blast
ultimately show ?thesis using * by blast
next
case 2
from x_hx have "x \ H''" ..
moreover have "y \ H''"
proof -
have "(y, h y) \ graph H' h'" by (rule y_hy)
also have "\ \ graph H'' h''" by fact
finally have "(y, h y) \ graph H'' h''" .
then show ?thesis ..
qed
moreover from u c'' have "graph H'' h'' \ graph H h" by blast
ultimately show ?thesis using ** by blast
qed
qed
text \<open>
\<^medskip>
The relation induced by the graph of the supremum of a chain \<open>c\<close> is
definite, i.e.\ it is the graph of a function.
\<close>
lemma sup_definite:
assumes M_def: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and xy: "(x, y) \ \c"
and xz: "(x, z) \ \c"
shows "z = y"
proof -
from cM have c: "c \ M" ..
from xy obtain G1 where xy': "(x, y) \ G1" and G1: "G1 \ c" ..
from xz obtain G2 where xz': "(x, z) \ G2" and G2: "G2 \ c" ..
from G1 c have "G1 \ M" ..
then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"
unfolding M_def by blast
from G2 c have "G2 \ M" ..
then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"
unfolding M_def by blast
txt \<open>\<open>G\<^sub>1\<close> is contained in \<open>G\<^sub>2\<close> or vice versa, since both \<open>G\<^sub>1\<close> and \<open>G\<^sub>2\<close>
are members of \<open>c\<close>. \label{cases2}\<close>
from cM G1 G2 consider "G1 \ G2" | "G2 \ G1"
by (blast dest: chainsD)
then show ?thesis
proof cases
case 1
with xy' G2_rep have "(x, y) \ graph H2 h2" by blast
then have "y = h2 x" ..
also
from xz' G2_rep have "(x, z) \ graph H2 h2" by (simp only:)
then have "z = h2 x" ..
finally show ?thesis .
next
case 2
with xz' G1_rep have "(x, z) \ graph H1 h1" by blast
then have "z = h1 x" ..
also
from xy' G1_rep have "(x, y) \ graph H1 h1" by (simp only:)
then have "y = h1 x" ..
finally show ?thesis ..
qed
qed
text \<open>
\<^medskip>
The limit function \<open>h\<close> is linear. Every element \<open>x\<close> in the domain of \<open>h\<close> is
in the domain of a function \<open>h'\<close> in the chain of norm preserving extensions.
Furthermore, \<open>h\<close> is an extension of \<open>h'\<close> so the function values of \<open>x\<close> are
identical for \<open>h'\<close> and \<open>h\<close>. Finally, the function \<open>h'\<close> is linear by
construction of \<open>M\<close>.
\<close>
lemma sup_lf:
assumes M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and u: "graph H h = \c"
shows "linearform H h"
proof
fix x y assume x: "x \ H" and y: "y \ H"
with M cM u obtain H' h' where
x': "x \ H'" and y': "y \ H'"
and b: "graph H' h' \ graph H h"
and linearform: "linearform H' h'"
and subspace: "H' \ E"
by (rule some_H'h'2 [elim_format]) blast
show "h (x + y) = h x + h y"
proof -
from linearform x' y' have "h' (x + y) = h' x + h' y"
by (rule linearform.add)
also from b x' have "h' x = h x" ..
also from b y' have "h' y = h y" ..
also from subspace x' y' have "x + y \ H'"
by (rule subspace.add_closed)
with b have "h' (x + y) = h (x + y)" ..
finally show ?thesis .
qed
next
fix x a assume x: "x \ H"
with M cM u obtain H' h' where
x': "x \ H'"
and b: "graph H' h' \ graph H h"
and linearform: "linearform H' h'"
and subspace: "H' \ E"
by (rule some_H'h' [elim_format]) blast
show "h (a \ x) = a * h x"
proof -
from linearform x' have "h' (a \<cdot> x) = a * h' x"
by (rule linearform.mult)
also from b x' have "h' x = h x" ..
also from subspace x' have "a \ x \ H'"
by (rule subspace.mult_closed)
with b have "h' (a \ x) = h (a \ x)" ..
finally show ?thesis .
qed
qed
text \<open>
\<^medskip>
The limit of a non-empty chain of norm preserving extensions of \<open>f\<close> is an
extension of \<open>f\<close>, since every element of the chain is an extension of \<open>f\<close>
and the supremum is an extension for every element of the chain.
\<close>
lemma sup_ext:
assumes graph: "graph H h = \c"
and M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and ex: "\x. x \ c"
shows "graph F f \ graph H h"
proof -
from ex obtain x where xc: "x \ c" ..
from cM have "c \ M" ..
with xc have "x \ M" ..
with M have "x \ norm_pres_extensions E p F f"
by (simp only:)
then obtain G g where "x = graph G g" and "graph F f \ graph G g" ..
then have "graph F f \ x" by (simp only:)
also from xc have "\ \ \c" by blast
also from graph have "\ = graph H h" ..
finally show ?thesis .
qed
text \<open>
\<^medskip>
The domain \<open>H\<close> of the limit function is a superspace of \<open>F\<close>, since \<open>F\<close> is a
subset of \<open>H\<close>. The existence of the \<open>0\<close> element in \<open>F\<close> and the closure
properties follow from the fact that \<open>F\<close> is a vector space.
\<close>
lemma sup_supF:
assumes graph: "graph H h = \c"
and M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and ex: "\x. x \ c"
and FE: "F \ E"
shows "F \ H"
proof
from FE show "F \ {}" by (rule subspace.non_empty)
from graph M cM ex have "graph F f \ graph H h" by (rule sup_ext)
then show "F \ H" ..
fix x y assume "x \ F" and "y \ F"
with FE show "x + y \ F" by (rule subspace.add_closed)
next
fix x a assume "x \ F"
with FE show "a \ x \ F" by (rule subspace.mult_closed)
qed
text \<open>
\<^medskip>
The domain \<open>H\<close> of the limit function is a subspace of \<open>E\<close>.
\<close>
lemma sup_subE:
assumes graph: "graph H h = \c"
and M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
and ex: "\x. x \ c"
and FE: "F \ E"
and E: "vectorspace E"
shows "H \ E"
proof
show "H \ {}"
proof -
from FE E have "0 \ F" by (rule subspace.zero)
also from graph M cM ex FE have "F \ H" by (rule sup_supF)
then have "F \ H" ..
finally show ?thesis by blast
qed
show "H \ E"
proof
fix x assume "x \ H"
with M cM graph
obtain H' where x: "x \ H'" and H'E: "H' \ E"
by (rule some_H'h' [elim_format]) blast
from H'E have "H' \<subseteq> E" ..
with x show "x \ E" ..
qed
fix x y assume x: "x \ H" and y: "y \ H"
show "x + y \ H"
proof -
from M cM graph x y obtain H' h' where
x': "x \ H'" and y': "y \ H'" and H'E: "H' \ E"
and graphs: "graph H' h' \ graph H h"
by (rule some_H'h'2 [elim_format]) blast
from H'E x' y' have "x + y \ H'"
by (rule subspace.add_closed)
also from graphs have "H' \ H" ..
finally show ?thesis .
qed
next
fix x a assume x: "x \ H"
show "a \ x \ H"
proof -
from M cM graph x
obtain H' h' where x': "x \ H'" and H'E: "H' \ E"
and graphs: "graph H' h' \ graph H h"
by (rule some_H'h' [elim_format]) blast
from H'E x' have "a \ x \ H'" by (rule subspace.mult_closed)
also from graphs have "H' \ H" ..
finally show ?thesis .
qed
qed
text \<open>
\<^medskip>
The limit function is bounded by the norm \<open>p\<close> as well, since all elements in
the chain are bounded by \<open>p\<close>.
\<close>
lemma sup_norm_pres:
assumes graph: "graph H h = \c"
and M: "M = norm_pres_extensions E p F f"
and cM: "c \ chains M"
shows "\x \ H. h x \ p x"
proof
fix x assume "x \ H"
with M cM graph obtain H' h' where x': "x \ H'"
and graphs: "graph H' h' \ graph H h"
and a: "\x \ H'. h' x \ p x"
by (rule some_H'h' [elim_format]) blast
from graphs x' have [symmetric]: "h' x = h x" ..
also from a x' have "h' x \<le> p x " ..
finally show "h x \ p x" .
qed
text \<open>
\<^medskip>
The following lemma is a property of linear forms on real vector spaces. It
will be used for the lemma \<open>abs_Hahn_Banach\<close> (see page
\pageref{abs-Hahn-Banach}). \label{abs-ineq-iff} For real vector spaces the
following inequality are equivalent:
\begin{center}
\begin{tabular}{lll}
\<open>\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x\<close> & and &
\<open>\<forall>x \<in> H. h x \<le> p x\<close> \\
\end{tabular}
\end{center}
\<close>
lemma abs_ineq_iff:
assumes "subspace H E" and "vectorspace E" and "seminorm E p"
and "linearform H h"
shows "(\x \ H. \h x\ \ p x) = (\x \ H. h x \ p x)" (is "?L = ?R")
proof
interpret subspace H E by fact
interpret vectorspace E by fact
interpret seminorm E p by fact
interpret linearform H h by fact
have H: "vectorspace H" using \<open>vectorspace E\<close> ..
{
assume l: ?L
show ?R
proof
fix x assume x: "x \ H"
have "h x \ \h x\" by arith
also from l x have "\ \ p x" ..
finally show "h x \ p x" .
qed
next
assume r: ?R
show ?L
proof
fix x assume x: "x \ H"
show "\b\ \ a" when "- a \ b" "b \ a" for a b :: real
using that by arith
from \<open>linearform H h\<close> and H x
have "- h x = h (- x)" by (rule linearform.neg [symmetric])
also
from H x have "- x \ H" by (rule vectorspace.neg_closed)
with r have "h (- x) \ p (- x)" ..
also have "\ = p x"
using \<open>seminorm E p\<close> \<open>vectorspace E\<close>
proof (rule seminorm.minus)
from x show "x \ E" ..
qed
finally have "- h x \ p x" .
then show "- p x \ h x" by simp
from r x show "h x \ p x" ..
qed
}
qed
end
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