(* Title: HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Extending non-maximal functions\<close>
theory Hahn_Banach_Ext_Lemmas
imports Function_Norm
begin
text \<open>
In this section the following context is presumed. Let \<open>E\<close> be a real vector
space with a seminorm \<open>q\<close> on \<open>E\<close>. \<open>F\<close> is a subspace of \<open>E\<close> and \<open>f\<close> a linear
function on \<open>F\<close>. We consider a subspace \<open>H\<close> of \<open>E\<close> that is a superspace of
\<open>F\<close> and a linear form \<open>h\<close> on \<open>H\<close>. \<open>H\<close> is a not equal to \<open>E\<close> and \<open>x\<^sub>0\<close> is an
element in \<open>E - H\<close>. \<open>H\<close> is extended to the direct sum \<open>H' = H + lin x\<^sub>0\<close>, so
for any \<open>x \<in> H'\<close> the decomposition of \<open>x = y + a \<cdot> x\<close> with \<open>y \<in> H\<close> is
unique. \<open>h'\<close> is defined on \<open>H'\<close> by \<open>h' x = h y + a \<cdot> \<xi>\<close> for a certain \<open>\<xi>\<close>.
Subsequently we show some properties of this extension \<open>h'\<close> of \<open>h\<close>.
\<^medskip>
This lemma will be used to show the existence of a linear extension of \<open>f\<close>
(see page \pageref{ex-xi-use}). It is a consequence of the completeness of
\<open>\<real>\<close>. To show
\begin{center}
\begin{tabular}{l}
\<open>\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y\<close>
\end{tabular}
\end{center}
\<^noindent> it suffices to show that
\begin{center}
\begin{tabular}{l}
\<open>\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v\<close>
\end{tabular}
\end{center}
\<close>
lemma ex_xi:
assumes "vectorspace F"
assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"
shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
proof -
interpret vectorspace F by fact
txt \<open>From the completeness of the reals follows:
The set \<open>S = {a u. u \<in> F}\<close> has a supremum, if it is
non-empty and has an upper bound.\<close>
let ?S = "{a u | u. u \<in> F}"
have "\<exists>xi. lub ?S xi"
proof (rule real_complete)
have "a 0 \<in> ?S" by blast
then show "\<exists>X. X \<in> ?S" ..
have "\<forall>y \<in> ?S. y \<le> b 0"
proof
fix y assume y: "y \<in> ?S"
then obtain u where u: "u \<in> F" and y: "y = a u" by blast
from u and zero have "a u \<le> b 0" by (rule r)
with y show "y \<le> b 0" by (simp only:)
qed
then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..
qed
then obtain xi where xi: "lub ?S xi" ..
{
fix y assume "y \<in> F"
then have "a y \<in> ?S" by blast
with xi have "a y \<le> xi" by (rule lub.upper)
}
moreover {
fix y assume y: "y \<in> F"
from xi have "xi \<le> b y"
proof (rule lub.least)
fix au assume "au \<in> ?S"
then obtain u where u: "u \<in> F" and au: "au = a u" by blast
from u y have "a u \<le> b y" by (rule r)
with au show "au \<le> b y" by (simp only:)
qed
}
ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast
qed
text \<open>
\<^medskip>
The function \<open>h'\<close> is defined as a \<open>h' x = h y + a \<cdot> \<xi>\<close> where \<open>x = y + a \<cdot> \<xi>\<close>
is a linear extension of \<open>h\<close> to \<open>H'\<close>.
\<close>
lemma h'_lf:
assumes h'_def: "\<And>x. h' x = (let (y, a) =
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"
and H'_def: "H' = H + lin x0"
and HE: "H \<unlhd> E"
assumes "linearform H h"
assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
assumes E: "vectorspace E"
shows "linearform H' h'"
proof -
interpret linearform H h by fact
interpret vectorspace E by fact
show ?thesis
proof
note E = \<open>vectorspace E\<close>
have H': "vectorspace H'"
proof (unfold H'_def)
from \<open>x0 \<in> E\<close>
have "lin x0 \<unlhd> E" ..
with HE show "vectorspace (H + lin x0)" using E ..
qed
{
fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
show "h' (x1 + x2) = h' x1 + h' x2"
proof -
from H' x1 x2 have "x1 + x2 \<in> H'"
by (rule vectorspace.add_closed)
with x1 x2 obtain y y1 y2 a a1 a2 where
x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"
and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"
unfolding H'_def sum_def lin_def by blast
have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0
proof (rule decomp_H') text_raw \<open>\label{decomp-H-use}\<close>
from HE y1 y2 show "y1 + y2 \<in> H"
by (rule subspace.add_closed)
from x0 and HE y y1 y2
have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto
with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"
by (simp add: add_ac add_mult_distrib2)
also note x1x2
finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .
qed
from h'_def x1x2 E HE y x0
have "h' (x1 + x2) = h y + a * xi"
by (rule h'_definite)
also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"
by (simp only: ya)
also from y1 y2 have "h (y1 + y2) = h y1 + h y2"
by simp
also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
by (simp add: distrib_right)
also from h'_def x1_rep E HE y1 x0
have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
also from h'_def x2_rep E HE y2 x0
have "h y2 + a2 * xi = h' x2"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
next
fix x1 c assume x1: "x1 \<in> H'"
show "h' (c \<cdot> x1) = c * (h' x1)"
proof -
from H' x1 have ax1: "c \<cdot> x1 \<in> H'"
by (rule vectorspace.mult_closed)
with x1 obtain y a y1 a1 where
cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"
and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
unfolding H'_def sum_def lin_def by blast
have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0
proof (rule decomp_H')
from HE y1 show "c \<cdot> y1 \<in> H"
by (rule subspace.mult_closed)
from x0 and HE y y1
have "x0 \<in> E" "y \<in> E" "y1 \<in> E" by auto
with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"
by (simp add: mult_assoc add_mult_distrib1)
also note cx1_rep
finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .
qed
from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"
by (rule h'_definite)
also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"
by (simp only: ya)
also from y1 have "h (c \<cdot> y1) = c * h y1"
by simp
also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"
by (simp only: distrib_left)
also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
}
qed
qed
text \<open>
\<^medskip>
The linear extension \<open>h'\<close> of \<open>h\<close> is bounded by the seminorm \<open>p\<close>.
\<close>
lemma h'_norm_pres:
assumes h'_def: "\<And>x. h' x = (let (y, a) =
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"
and H'_def: "H' = H + lin x0"
and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
assumes E: "vectorspace E" and HE: "subspace H E"
and "seminorm E p" and "linearform H h"
assumes a: "\<forall>y \<in> H. h y \<le> p y"
and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"
shows "\<forall>x \<in> H'. h' x \<le> p x"
proof -
interpret vectorspace E by fact
interpret subspace H E by fact
interpret seminorm E p by fact
interpret linearform H h by fact
show ?thesis
proof
fix x assume x': "x \<in> H'"
show "h' x \<le> p x"
proof -
from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto
from x' obtain y a where
x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"
unfolding H'_def sum_def lin_def by blast
from y have y': "y \<in> E" ..
from y have ay: "inverse a \<cdot> y \<in> H" by simp
from h'_def x_rep E HE y x0 have "h' x = h y + a * xi"
by (rule h'_definite)
also have "\<dots> \<le> p (y + a \<cdot> x0)"
proof (rule linorder_cases)
assume z: "a = 0"
then have "h y + a * xi = h y" by simp
also from a y have "\<dots> \<le> p y" ..
also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp
finally show ?thesis .
next
txt \<open>In the case \<open>a < 0\<close>, we use \<open>a\<^sub>1\<close>
with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>
assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp
from a1 ay
have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..
with lz have "a * xi \<le>
a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
by (simp add: mult_left_mono_neg order_less_imp_le)
also have "\<dots> =
- a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
by (simp add: right_diff_distrib)
also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =
p (a \<cdot> (inverse a \<cdot> y + x0))"
by (simp add: abs_homogenous)
also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
by (simp add: add_mult_distrib1 mult_assoc [symmetric])
also from nz y have "a * (h (inverse a \<cdot> y)) = h y"
by simp
finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
then show ?thesis by simp
next
txt \<open>In the case \<open>a > 0\<close>, we use \<open>a\<^sub>2\<close>
with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>
assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp
from a2 ay
have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..
with gz have "a * xi \<le>
a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
by simp
also have "\<dots> = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"
by (simp add: right_diff_distrib)
also from gz x0 y'
have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"
by (simp add: abs_homogenous)
also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
by (simp add: add_mult_distrib1 mult_assoc [symmetric])
also from nz y have "a * h (inverse a \<cdot> y) = h y"
by simp
finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
then show ?thesis by simp
qed
also from x_rep have "\<dots> = p x" by (simp only:)
finally show ?thesis .
qed
qed
qed
end