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(* Title: HOL/Real/HahnBanach/HahnBanachSupLemmas.thy
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ID: $Id$
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Author: Gertrud Bauer, TU Munich
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*)
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header {* The supremum w.r.t.~the function order *}
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theory HahnBanachSupLemmas = FunctionNorm + ZornLemma:
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text {*
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This section contains some lemmas that will be used in the proof of
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the Hahn-Banach Theorem. In this section the following context is
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presumed. Let @{text E} be a real vector space with a seminorm
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@{text p} on @{text E}. @{text F} is a subspace of @{text E} and
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@{text f} a linear form on @{text F}. We consider a chain @{text c}
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of norm-preserving extensions of @{text f}, such that @{text "\<Union>c =
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graph H h"}. We will show some properties about the limit function
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@{text h}, i.e.\ the supremum of the chain @{text c}.
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\medskip Let @{text c} be a chain of norm-preserving extensions of
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the function @{text f} and let @{text "graph H h"} be the supremum
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of @{text c}. Every element in @{text H} is member of one of the
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elements of the chain.
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*}
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lemma some_H'h't:
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assumes M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and u: "graph H h = \<Union>c"
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and x: "x \<in> H"
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shows "\<exists>H' h'. graph H' h' \<in> c
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\<and> (x, h x) \<in> graph H' h'
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\<and> linearform H' h' \<and> H' \<unlhd> E
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\<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h'
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\<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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from x have "(x, h x) \<in> graph H h" ..
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also from u have "\<dots> = \<Union>c" .
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finally obtain g where gc: "g \<in> c" and gh: "(x, h x) \<in> g" by blast
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from cM have "c \<subseteq> M" ..
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with gc have "g \<in> M" ..
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also from M have "\<dots> = norm_pres_extensions E p F f" .
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finally obtain H' and h' where g: "g = graph H' h'"
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and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"
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"graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x" ..
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from gc and g have "graph H' h' \<in> c" by (simp only:)
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moreover from gh and g have "(x, h x) \<in> graph H' h'" by (simp only:)
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ultimately show ?thesis using * by blast
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qed
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text {*
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\medskip Let @{text c} be a chain of norm-preserving extensions of
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the function @{text f} and let @{text "graph H h"} be the supremum
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of @{text c}. Every element in the domain @{text H} of the supremum
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function is member of the domain @{text H'} of some function @{text
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h'}, such that @{text h} extends @{text h'}.
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*}
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lemma some_H'h':
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assumes M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and u: "graph H h = \<Union>c"
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and x: "x \<in> H"
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shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
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\<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
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\<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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from M cM u x obtain H' h' where
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x_hx: "(x, h x) \<in> graph H' h'"
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and c: "graph H' h' \<in> c"
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and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"
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"graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"
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by (rule some_H'h't [elim_format]) blast
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from x_hx have "x \<in> H'" ..
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moreover from cM u c have "graph H' h' \<subseteq> graph H h"
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by (simp only: chain_ball_Union_upper)
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ultimately show ?thesis using * by blast
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qed
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text {*
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\medskip Any two elements @{text x} and @{text y} in the domain
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@{text H} of the supremum function @{text h} are both in the domain
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@{text H'} of some function @{text h'}, such that @{text h} extends
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@{text h'}.
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*}
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lemma some_H'h'2:
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assumes M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and u: "graph H h = \<Union>c"
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and x: "x \<in> H"
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and y: "y \<in> H"
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shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H'
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\<and> graph H' h' \<subseteq> graph H h
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\<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'
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\<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
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proof -
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txt {* @{text y} is in the domain @{text H''} of some function @{text h''},
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such that @{text h} extends @{text h''}. *}
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from M cM u and y obtain H' h' where
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y_hy: "(y, h y) \<in> graph H' h'"
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and c': "graph H' h' \<in> c"
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and * :
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"linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"
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"graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"
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by (rule some_H'h't [elim_format]) blast
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txt {* @{text x} is in the domain @{text H'} of some function @{text h'},
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such that @{text h} extends @{text h'}. *}
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from M cM u and x obtain H'' h'' where
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x_hx: "(x, h x) \<in> graph H'' h''"
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and c'': "graph H'' h'' \<in> c"
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and ** :
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"linearform H'' h''" "H'' \<unlhd> E" "F \<unlhd> H''"
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"graph F f \<subseteq> graph H'' h''" "\<forall>x \<in> H''. h'' x \<le> p x"
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by (rule some_H'h't [elim_format]) blast
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txt {* Since both @{text h'} and @{text h''} are elements of the chain,
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@{text h''} is an extension of @{text h'} or vice versa. Thus both
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@{text x} and @{text y} are contained in the greater
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one. \label{cases1}*}
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from cM have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''"
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(is "?case1 \<or> ?case2") ..
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then show ?thesis
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proof
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assume ?case1
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have "(x, h x) \<in> graph H'' h''" .
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also have "... \<subseteq> graph H' h'" .
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finally have xh:"(x, h x) \<in> graph H' h'" .
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then have "x \<in> H'" ..
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moreover from y_hy have "y \<in> H'" ..
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moreover from cM u and c' have "graph H' h' \<subseteq> graph H h"
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by (simp only: chain_ball_Union_upper)
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ultimately show ?thesis using * by blast
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next
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assume ?case2
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from x_hx have "x \<in> H''" ..
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moreover {
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from y_hy have "(y, h y) \<in> graph H' h'" .
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also have "\<dots> \<subseteq> graph H'' h''" .
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finally have "(y, h y) \<in> graph H'' h''" .
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} then have "y \<in> H''" ..
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moreover from cM u and c'' have "graph H'' h'' \<subseteq> graph H h"
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by (simp only: chain_ball_Union_upper)
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ultimately show ?thesis using ** by blast
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qed
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qed
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text {*
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\medskip The relation induced by the graph of the supremum of a
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chain @{text c} is definite, i.~e.~t is the graph of a function.
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*}
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lemma sup_definite:
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assumes M_def: "M \<equiv> norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and xy: "(x, y) \<in> \<Union>c"
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and xz: "(x, z) \<in> \<Union>c"
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shows "z = y"
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proof -
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from cM have c: "c \<subseteq> M" ..
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from xy obtain G1 where xy': "(x, y) \<in> G1" and G1: "G1 \<in> c" ..
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from xz obtain G2 where xz': "(x, z) \<in> G2" and G2: "G2 \<in> c" ..
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from G1 c have "G1 \<in> M" ..
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then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"
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by (unfold M_def) blast
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from G2 c have "G2 \<in> M" ..
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then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"
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by (unfold M_def) blast
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txt {* @{text "G\<^sub>1"} is contained in @{text "G\<^sub>2"}
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or vice versa, since both @{text "G\<^sub>1"} and @{text
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"G\<^sub>2"} are members of @{text c}. \label{cases2}*}
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from cM G1 G2 have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") ..
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then show ?thesis
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proof
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assume ?case1
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with xy' G2_rep have "(x, y) \<in> graph H2 h2" by blast
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hence "y = h2 x" ..
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also
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from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:)
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hence "z = h2 x" ..
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finally show ?thesis .
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next
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assume ?case2
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with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast
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hence "z = h1 x" ..
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also
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from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:)
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hence "y = h1 x" ..
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finally show ?thesis ..
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qed
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qed
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text {*
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\medskip The limit function @{text h} is linear. Every element
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@{text x} in the domain of @{text h} is in the domain of a function
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@{text h'} in the chain of norm preserving extensions. Furthermore,
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@{text h} is an extension of @{text h'} so the function values of
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@{text x} are identical for @{text h'} and @{text h}. Finally, the
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function @{text h'} is linear by construction of @{text M}.
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*}
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lemma sup_lf:
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assumes M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and u: "graph H h = \<Union>c"
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shows "linearform H h"
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proof
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fix x y assume x: "x \<in> H" and y: "y \<in> H"
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with M cM u obtain H' h' where
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x': "x \<in> H'" and y': "y \<in> H'"
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and b: "graph H' h' \<subseteq> graph H h"
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and linearform: "linearform H' h'"
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and subspace: "H' \<unlhd> E"
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by (rule some_H'h'2 [elim_format]) blast
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show "h (x + y) = h x + h y"
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proof -
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from linearform x' y' have "h' (x + y) = h' x + h' y"
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by (rule linearform.add)
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also from b x' have "h' x = h x" ..
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also from b y' have "h' y = h y" ..
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also from subspace x' y' have "x + y \<in> H'"
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by (rule subspace.add_closed)
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with b have "h' (x + y) = h (x + y)" ..
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finally show ?thesis .
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qed
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next
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fix x a assume x: "x \<in> H"
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with M cM u obtain H' h' where
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x': "x \<in> H'"
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and b: "graph H' h' \<subseteq> graph H h"
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and linearform: "linearform H' h'"
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and subspace: "H' \<unlhd> E"
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by (rule some_H'h' [elim_format]) blast
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show "h (a \<cdot> x) = a * h x"
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proof -
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from linearform x' have "h' (a \<cdot> x) = a * h' x"
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by (rule linearform.mult)
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also from b x' have "h' x = h x" ..
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also from subspace x' have "a \<cdot> x \<in> H'"
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by (rule subspace.mult_closed)
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with b have "h' (a \<cdot> x) = h (a \<cdot> x)" ..
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finally show ?thesis .
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qed
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qed
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text {*
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\medskip The limit of a non-empty chain of norm preserving
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extensions of @{text f} is an extension of @{text f}, since every
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element of the chain is an extension of @{text f} and the supremum
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is an extension for every element of the chain.
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*}
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lemma sup_ext:
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assumes graph: "graph H h = \<Union>c"
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and M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and ex: "\<exists>x. x \<in> c"
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shows "graph F f \<subseteq> graph H h"
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proof -
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from ex obtain x where xc: "x \<in> c" ..
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from cM have "c \<subseteq> M" ..
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with xc have "x \<in> M" ..
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with M have "x \<in> norm_pres_extensions E p F f"
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by (simp only:)
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then obtain G g where "x = graph G g" and "graph F f \<subseteq> graph G g" ..
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then have "graph F f \<subseteq> x" by (simp only:)
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also from xc have "\<dots> \<subseteq> \<Union>c" by blast
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also from graph have "\<dots> = graph H h" ..
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finally show ?thesis .
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qed
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text {*
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\medskip The domain @{text H} of the limit function is a superspace
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of @{text F}, since @{text F} is a subset of @{text H}. The
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existence of the @{text 0} element in @{text F} and the closure
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properties follow from the fact that @{text F} is a vector space.
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*}
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lemma sup_supF:
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assumes graph: "graph H h = \<Union>c"
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and M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and ex: "\<exists>x. x \<in> c"
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and FE: "F \<unlhd> E"
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shows "F \<unlhd> H"
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proof
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from FE show "F \<noteq> {}" by (rule subspace.non_empty)
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from graph M cM ex have "graph F f \<subseteq> graph H h" by (rule sup_ext)
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then show "F \<subseteq> H" ..
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fix x y assume "x \<in> F" and "y \<in> F"
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with FE show "x + y \<in> F" by (rule subspace.add_closed)
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next
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fix x a assume "x \<in> F"
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with FE show "a \<cdot> x \<in> F" by (rule subspace.mult_closed)
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qed
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text {*
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\medskip The domain @{text H} of the limit function is a subspace of
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@{text E}.
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*}
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lemma sup_subE:
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assumes graph: "graph H h = \<Union>c"
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and M: "M = norm_pres_extensions E p F f"
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and cM: "c \<in> chain M"
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and ex: "\<exists>x. x \<in> c"
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and FE: "F \<unlhd> E"
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and E: "vectorspace E"
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shows "H \<unlhd> E"
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proof
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323 |
show "H \<noteq> {}"
|
|
324 |
proof -
|
|
325 |
from FE E have "0 \<in> F" by (rule subvectorspace.zero)
|
|
326 |
also from graph M cM ex FE have "F \<unlhd> H" by (rule sup_supF)
|
|
327 |
then have "F \<subseteq> H" ..
|
|
328 |
finally show ?thesis by blast
|
|
329 |
qed
|
|
330 |
show "H \<subseteq> E"
|
9261
|
331 |
proof
|
13515
|
332 |
fix x assume "x \<in> H"
|
|
333 |
with M cM graph
|
|
334 |
obtain H' h' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"
|
|
335 |
by (rule some_H'h' [elim_format]) blast
|
|
336 |
from H'E have "H' \<subseteq> E" ..
|
|
337 |
with x show "x \<in> E" ..
|
|
338 |
qed
|
|
339 |
fix x y assume x: "x \<in> H" and y: "y \<in> H"
|
|
340 |
show "x + y \<in> H"
|
|
341 |
proof -
|
|
342 |
from M cM graph x y obtain H' h' where
|
|
343 |
x': "x \<in> H'" and y': "y \<in> H'" and H'E: "H' \<unlhd> E"
|
|
344 |
and graphs: "graph H' h' \<subseteq> graph H h"
|
|
345 |
by (rule some_H'h'2 [elim_format]) blast
|
|
346 |
from H'E x' y' have "x + y \<in> H'"
|
|
347 |
by (rule subspace.add_closed)
|
|
348 |
also from graphs have "H' \<subseteq> H" ..
|
|
349 |
finally show ?thesis .
|
|
350 |
qed
|
|
351 |
next
|
|
352 |
fix x a assume x: "x \<in> H"
|
|
353 |
show "a \<cdot> x \<in> H"
|
|
354 |
proof -
|
|
355 |
from M cM graph x
|
|
356 |
obtain H' h' where x': "x \<in> H'" and H'E: "H' \<unlhd> E"
|
|
357 |
and graphs: "graph H' h' \<subseteq> graph H h"
|
|
358 |
by (rule some_H'h' [elim_format]) blast
|
|
359 |
from H'E x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed)
|
|
360 |
also from graphs have "H' \<subseteq> H" ..
|
|
361 |
finally show ?thesis .
|
9261
|
362 |
qed
|
|
363 |
qed
|
7917
|
364 |
|
10687
|
365 |
text {*
|
|
366 |
\medskip The limit function is bounded by the norm @{text p} as
|
|
367 |
well, since all elements in the chain are bounded by @{text p}.
|
9261
|
368 |
*}
|
7917
|
369 |
|
9374
|
370 |
lemma sup_norm_pres:
|
13515
|
371 |
assumes graph: "graph H h = \<Union>c"
|
|
372 |
and M: "M = norm_pres_extensions E p F f"
|
|
373 |
and cM: "c \<in> chain M"
|
|
374 |
shows "\<forall>x \<in> H. h x \<le> p x"
|
9261
|
375 |
proof
|
9503
|
376 |
fix x assume "x \<in> H"
|
13515
|
377 |
with M cM graph obtain H' h' where x': "x \<in> H'"
|
|
378 |
and graphs: "graph H' h' \<subseteq> graph H h"
|
10687
|
379 |
and a: "\<forall>x \<in> H'. h' x \<le> p x"
|
13515
|
380 |
by (rule some_H'h' [elim_format]) blast
|
|
381 |
from graphs x' have [symmetric]: "h' x = h x" ..
|
|
382 |
also from a x' have "h' x \<le> p x " ..
|
|
383 |
finally show "h x \<le> p x" .
|
9261
|
384 |
qed
|
7917
|
385 |
|
10687
|
386 |
text {*
|
|
387 |
\medskip The following lemma is a property of linear forms on real
|
|
388 |
vector spaces. It will be used for the lemma @{text abs_HahnBanach}
|
|
389 |
(see page \pageref{abs-HahnBanach}). \label{abs-ineq-iff} For real
|
|
390 |
vector spaces the following inequations are equivalent:
|
|
391 |
\begin{center}
|
|
392 |
\begin{tabular}{lll}
|
|
393 |
@{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
|
|
394 |
@{text "\<forall>x \<in> H. h x \<le> p x"} \\
|
|
395 |
\end{tabular}
|
|
396 |
\end{center}
|
9261
|
397 |
*}
|
7917
|
398 |
|
10687
|
399 |
lemma abs_ineq_iff:
|
13515
|
400 |
includes subvectorspace H E + seminorm E p + linearform H h
|
|
401 |
shows "(\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)" (is "?L = ?R")
|
|
402 |
proof
|
|
403 |
have h: "vectorspace H" by (rule vectorspace)
|
|
404 |
{
|
9261
|
405 |
assume l: ?L
|
|
406 |
show ?R
|
|
407 |
proof
|
9503
|
408 |
fix x assume x: "x \<in> H"
|
13515
|
409 |
have "h x \<le> \<bar>h x\<bar>" by arith
|
|
410 |
also from l x have "\<dots> \<le> p x" ..
|
10687
|
411 |
finally show "h x \<le> p x" .
|
9261
|
412 |
qed
|
|
413 |
next
|
|
414 |
assume r: ?R
|
|
415 |
show ?L
|
10687
|
416 |
proof
|
13515
|
417 |
fix x assume x: "x \<in> H"
|
10687
|
418 |
show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a"
|
9261
|
419 |
by arith
|
10687
|
420 |
show "- p x \<le> h x"
|
9261
|
421 |
proof (rule real_minus_le)
|
13515
|
422 |
have "linearform H h" .
|
|
423 |
from h this x have "- h x = h (- x)"
|
|
424 |
by (rule vectorspace_linearform.neg [symmetric])
|
|
425 |
also from r x have "\<dots> \<le> p (- x)" by simp
|
|
426 |
also have "\<dots> = p x"
|
|
427 |
proof (rule seminorm_vectorspace.minus)
|
9503
|
428 |
show "x \<in> E" ..
|
9379
|
429 |
qed
|
10687
|
430 |
finally show "- h x \<le> p x" .
|
9261
|
431 |
qed
|
13515
|
432 |
from r x show "h x \<le> p x" ..
|
9261
|
433 |
qed
|
13515
|
434 |
}
|
10687
|
435 |
qed
|
7917
|
436 |
|
10687
|
437 |
end
|