--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Sun Jul 16 21:00:32 2000 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Mon Jul 17 13:58:18 2000 +0200
@@ -1,123 +1,182 @@
-theory HahnBanach = HahnBanachLemmas: text_raw {* \smallskip\\ *} (* from ~/Pub/TYPES99/HB/HahnBanach.thy *)
+(* Title: HOL/Real/HahnBanach/HahnBanach.thy
+ ID: $Id$
+ Author: Gertrud Bauer, TU Munich
+*)
+
+header {* The Hahn-Banach Theorem *}
+
+theory HahnBanach = HahnBanachLemmas:
+
+text {*
+ We present the proof of two different versions of the Hahn-Banach
+ Theorem, closely following \cite[\S36]{Heuser:1986}.
+*}
+
+subsection {* The Hahn-Banach Theorem for vector spaces *}
+
+text {*
+{\bf Hahn-Banach Theorem.}\quad
+ Let $F$ be a subspace of a real vector space $E$, let $p$ be a semi-norm on
+ $E$, and $f$ be a linear form defined on $F$ such that $f$ is bounded by
+ $p$, i.e. $\All {x\in F} f\ap x \leq p\ap x$. Then $f$ can be extended to
+ a linear form $h$ on $E$ such that $h$ is norm-preserving, i.e. $h$ is also
+ bounded by $p$.
+
+\bigskip
+{\bf Proof Sketch.}
+\begin{enumerate}
+\item Define $M$ as the set of norm-preserving extensions of $f$ to subspaces
+ of $E$. The linear forms in $M$ are ordered by domain extension.
+\item We show that every non-empty chain in $M$ has an upper bound in $M$.
+\item With Zorn's Lemma we conclude that there is a maximal function $g$ in
+ $M$.
+\item The domain $H$ of $g$ is the whole space $E$, as shown by classical
+ contradiction:
+\begin{itemize}
+\item Assuming $g$ is not defined on whole $E$, it can still be extended in a
+ norm-preserving way to a super-space $H'$ of $H$.
+\item Thus $g$ can not be maximal. Contradiction!
+\end{itemize}
+\end{enumerate}
+\bigskip
+*}
+
+(*
+text {* {\bf Theorem.} Let $f$ be a linear form defined on a subspace
+ $F$ of a real vector space $E$, such that $f$ is bounded by a seminorm
+ $p$.
+
+ Then $f$ can be extended to a linear form $h$ on $E$ that is again
+ bounded by $p$.
+
+ \bigskip{\bf Proof Outline.}
+ First we define the set $M$ of all norm-preserving extensions of $f$.
+ We show that every chain in $M$ has an upper bound in $M$.
+ With Zorn's lemma we can conclude that $M$ has a maximal element $g$.
+ We further show by contradiction that the domain $H$ of $g$ is the whole
+ vector space $E$.
+ If $H \neq E$, then $g$ can be extended in
+ a norm-preserving way to a greater vector space $H_0$.
+ So $g$ cannot be maximal in $M$.
+ \bigskip
+*}
+*)
theorem HahnBanach:
- "is_vectorspace E \\<Longrightarrow> is_subspace F E \\<Longrightarrow> is_seminorm E p
- \\<Longrightarrow> is_linearform F f \\<Longrightarrow> \\<forall>x \\<in> F. f x \\<le> p x
- \\<Longrightarrow> \\<exists>h. is_linearform E h \\<and> (\\<forall>x \\<in> F. h x = f x) \\<and> (\\<forall>x \\<in> E. h x \\<le> p x)"
+ "[| is_vectorspace E; is_subspace F E; is_seminorm E p;
+ is_linearform F f; \<forall>x \<in> F. f x \<le> p x |]
+ ==> \<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)"
-- {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$, *}
-- {* and $f$ a linear form on $F$ such that $f$ is bounded by $p$, *}
-- {* then $f$ can be extended to a linear form $h$ on $E$ in a norm-preserving way. \skp *}
proof -
assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
- and "is_linearform F f" "\\<forall>x \\<in> F. f x \\<le> p x"
+ and "is_linearform F f" "\<forall>x \<in> F. f x \<le> p x"
-- {* Assume the context of the theorem. \skp *}
def M == "norm_pres_extensions E p F f"
-- {* Define $M$ as the set of all norm-preserving extensions of $F$. \skp *}
{
- fix c assume "c \\<in> chain M" "\\<exists>x. x \\<in> c"
- have "\\<Union>c \\<in> M"
- txt {* Show that every non-empty chain $c$ of $M$ has an upper bound in $M$: *}
- txt {* $\Union c$ is greater than any element of the chain $c$, so it suffices to show $\Union c \in M$. *}
+ fix c assume "c \<in> chain M" "\<exists>x. x \<in> c"
+ have "\<Union> c \<in> M"
+ -- {* Show that every non-empty chain $c$ of $M$ has an upper bound in $M$: *}
+ -- {* $\Union c$ is greater than any element of the chain $c$, so it suffices to show $\Union c \in M$. *}
proof (unfold M_def, rule norm_pres_extensionI)
- show "\\<exists> H h. graph H h = \\<Union> c
- & is_linearform H h
- & is_subspace H E
- & is_subspace F H
- & graph F f \\<subseteq> graph H h
- & (\\<forall> x \\<in> H. h x \\<le> p x)"
+ show "\<exists>H h. graph H h = \<Union> c
+ \<and> is_linearform H h
+ \<and> is_subspace H E
+ \<and> is_subspace F H
+ \<and> graph F f \<subseteq> graph H h
+ \<and> (\<forall>x \<in> H. h x \<le> p x)"
proof (intro exI conjI)
- let ?H = "domain (\\<Union> c)"
- let ?h = "funct (\\<Union> c)"
+ let ?H = "domain (\<Union> c)"
+ let ?h = "funct (\<Union> c)"
- show a: "graph ?H ?h = \\<Union> c"
+ show a: "graph ?H ?h = \<Union> c"
proof (rule graph_domain_funct)
- fix x y z assume "(x, y) \\<in> \\<Union> c" "(x, z) \\<in> \\<Union> c"
+ fix x y z assume "(x, y) \<in> \<Union> c" "(x, z) \<in> \<Union> c"
show "z = y" by (rule sup_definite)
qed
show "is_linearform ?H ?h"
by (simp! add: sup_lf a)
- show "is_subspace ?H E" thm sup_subE [OF _ _ _ a]
- by (rule sup_subE [OF _ _ _ a]) (simp !)+
- (* FIXME by (rule sup_subE, rule a) (simp!)+; *)
+ show "is_subspace ?H E"
+ by (rule sup_subE, rule a) (simp!)+
show "is_subspace F ?H"
- by (rule sup_supF [OF _ _ _ a]) (simp!)+
- (* FIXME by (rule sup_supF, rule a) (simp!)+ *)
- show "graph F f \\<subseteq> graph ?H ?h"
- by (rule sup_ext [OF _ _ _ a]) (simp!)+
- (* FIXME by (rule sup_ext, rule a) (simp!)+*)
- show "\\<forall>x \\<in> ?H. ?h x \\<le> p x"
- by (rule sup_norm_pres [OF _ _ a]) (simp!)+
- (* FIXME by (rule sup_norm_pres, rule a) (simp!)+ *)
+ by (rule sup_supF, rule a) (simp!)+
+ show "graph F f \<subseteq> graph ?H ?h"
+ by (rule sup_ext, rule a) (simp!)+
+ show "\<forall>x \<in> ?H. ?h x \<le> p x"
+ by (rule sup_norm_pres, rule a) (simp!)+
qed
qed
}
- hence "\\<exists>g \\<in> M. \\<forall>x \\<in> M. g \\<subseteq> x \\<longrightarrow> g = x"
- txt {* With Zorn's Lemma we can conclude that there is a maximal element in $M$.\skp *}
+ hence "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x --> g = x"
+ -- {* With Zorn's Lemma we can conclude that there is a maximal element in $M$.\skp *}
proof (rule Zorn's_Lemma)
- txt {* We show that $M$ is non-empty: *}
- have "graph F f \\<in> norm_pres_extensions E p F f"
+ -- {* We show that $M$ is non-empty: *}
+ have "graph F f \<in> norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
have "is_vectorspace F" ..
thus "is_subspace F F" ..
qed (blast!)+
- thus "graph F f \\<in> M" by (simp!)
+ thus "graph F f \<in> M" by (simp!)
qed
thus ?thesis
proof
- fix g assume "g \\<in> M" "\\<forall>x \\<in> M. g \\<subseteq> x \\<longrightarrow> g = x"
+ fix g assume "g \<in> M" "\<forall>x \<in> M. g \<subseteq> x --> g = x"
-- {* We consider such a maximal element $g \in M$. \skp *}
show ?thesis
obtain H h where "graph H h = g" "is_linearform H h"
- "is_subspace H E" "is_subspace F H" "graph F f \\<subseteq> graph H h"
- "\\<forall>x \\<in> H. h x \\<le> p x"
- txt {* $g$ is a norm-preserving extension of $f$, in other words: *}
- txt {* $g$ is the graph of some linear form $h$ defined on a subspace $H$ of $E$, *}
- txt {* and $h$ is an extension of $f$ that is again bounded by $p$. \skp *}
+ "is_subspace H E" "is_subspace F H" "graph F f \<subseteq> graph H h"
+ "\<forall>x \<in> H. h x \<le> p x"
+ -- {* $g$ is a norm-preserving extension of $f$, in other words: *}
+ -- {* $g$ is the graph of some linear form $h$ defined on a subspace $H$ of $E$, *}
+ -- {* and $h$ is an extension of $f$ that is again bounded by $p$. \skp *}
proof -
- have "\\<exists> H h. graph H h = g & is_linearform H h
- & is_subspace H E & is_subspace F H
- & graph F f \\<subseteq> graph H h
- & (\\<forall>x \\<in> H. h x \\<le> p x)" by (simp! add: norm_pres_extension_D)
+ have "\<exists> H h. graph H h = g \<and> is_linearform H h
+ \<and> is_subspace H E \<and> is_subspace F H
+ \<and> graph F f \<subseteq> graph H h
+ \<and> (\<forall>x \<in> H. h x \<le> p x)" by (simp! add: norm_pres_extension_D)
thus ?thesis by (elim exE conjE) rule
qed
have h: "is_vectorspace H" ..
have "H = E"
-- {* We show that $h$ is defined on whole $E$ by classical contradiction. \skp *}
proof (rule classical)
- assume "H \\<noteq> E"
+ assume "H \<noteq> E"
-- {* Assume $h$ is not defined on whole $E$. Then show that $h$ can be extended *}
-- {* in a norm-preserving way to a function $h'$ with the graph $g'$. \skp *}
- have "\\<exists>g' \\<in> M. g \\<subseteq> g' \\<and> g \\<noteq> g'"
- obtain x' where "x' \\<in> E" "x' \\<notin> H"
- txt {* Pick $x' \in E \setminus H$. \skp *}
+ have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
+ obtain x' where "x' \<in> E" "x' \<notin> H"
+ -- {* Pick $x' \in E \setminus H$. \skp *}
proof -
- have "\\<exists>x' \\<in> E. x' \\<notin> H"
+ have "\<exists>x' \<in> E. x' \<notin> H"
proof (rule set_less_imp_diff_not_empty)
- have "H \\<subseteq> E" ..
- thus "H \\<subset> E" ..
+ have "H \<subseteq> E" ..
+ thus "H \<subset> E" ..
qed
thus ?thesis by blast
qed
- have x': "x' \\<noteq> \<zero>"
+ have x': "x' \<noteq> 0"
proof (rule classical)
- presume "x' = \<zero>"
- with h have "x' \\<in> H" by simp
+ presume "x' = 0"
+ with h have "x' \<in> H" by simp
thus ?thesis by contradiction
qed blast
def H' == "H + lin x'"
-- {* Define $H'$ as the direct sum of $H$ and the linear closure of $x'$. \skp *}
show ?thesis
- obtain xi where "\\<forall>y \\<in> H. - p (y + x') - h y \\<le> xi
- \\<and> xi \\<le> p (y + x') - h y"
- txt {* Pick a real number $\xi$ that fulfills certain inequations; this will *}
- txt {* be used to establish that $h'$ is a norm-preserving extension of $h$. \skp *}
+ obtain xi where "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
+ \<and> xi \<le> p (y + x') - h y"
+ -- {* Pick a real number $\xi$ that fulfills certain inequations; this will *}
+ -- {* be used to establish that $h'$ is a norm-preserving extension of $h$.
+ \label{ex-xi-use}\skp *}
proof -
- from h have "EX xi. ALL y:H. - p (y + x') - h y <= xi
- & xi <= p (y + x') - h y"
+ from h have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y <= xi
+ \<and> xi <= p (y + x') - h y"
proof (rule ex_xi)
- fix u v assume "u:H" "v:H"
+ fix u v assume "u \<in> H" "v \<in> H"
from h have "h v - h u = h (v - u)"
by (simp! add: linearform_diff)
also have "... <= p (v - u)"
@@ -138,25 +197,25 @@
thus ?thesis by rule rule
qed
- def h' == "\\<lambda>x. let (y,a) = \\<epsilon>(y,a). x = y + a \<prod> x' \\<and> y \\<in> H
+ def h' == "\<lambda>x. let (y,a) = SOME (y,a). x = y + a \<cdot> x' \<and> y \<in> H
in (h y) + a * xi"
-- {* Define the extension $h'$ of $h$ to $H'$ using $\xi$. \skp *}
show ?thesis
proof
- show "g \\<subseteq> graph H' h' \\<and> g \\<noteq> graph H' h'"
- txt {* Show that $h'$ is an extension of $h$ \dots \skp *}
-proof
- show "g \\<subseteq> graph H' h'"
+ show "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
+ -- {* Show that $h'$ is an extension of $h$ \dots \skp *}
+ proof
+ show "g \<subseteq> graph H' h'"
proof -
- have "graph H h \\<subseteq> graph H' h'"
+ have "graph H h \<subseteq> graph H' h'"
proof (rule graph_extI)
- fix t assume "t \\<in> H"
- have "(SOME (y, a). t = y + a \<prod> x' & y \\<in> H)
- = (t, #0)"
- by (rule decomp_H0_H [OF _ _ _ _ _ x'])
+ fix t assume "t \<in> H"
+ have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H)
+ = (t, #0)"
+ by (rule decomp_H'_H) (assumption+, rule x');
thus "h t = h' t" by (simp! add: Let_def)
next
- show "H \\<subseteq> H'"
+ show "H \<subseteq> H'"
proof (rule subspace_subset)
show "is_subspace H H'"
proof (unfold H'_def, rule subspace_vs_sum1)
@@ -167,33 +226,33 @@
qed
thus ?thesis by (simp!)
qed
- show "g \\<noteq> graph H' h'"
+ show "g \<noteq> graph H' h'"
proof -
- have "graph H h \\<noteq> graph H' h'"
+ have "graph H h \<noteq> graph H' h'"
proof
assume e: "graph H h = graph H' h'"
- have "x' \\<in> H'"
+ have "x' \<in> H'"
proof (unfold H'_def, rule vs_sumI)
- show "x' = \<zero> + x'" by (simp!)
- from h show "\<zero> \\<in> H" ..
- show "x' \\<in> lin x'" by (rule x_lin_x)
+ show "x' = 0 + x'" by (simp!)
+ from h show "0 \<in> H" ..
+ show "x' \<in> lin x'" by (rule x_lin_x)
qed
- hence "(x', h' x') \\<in> graph H' h'" ..
- with e have "(x', h' x') \\<in> graph H h" by simp
- hence "x' \\<in> H" ..
+ hence "(x', h' x') \<in> graph H' h'" ..
+ with e have "(x', h' x') \<in> graph H h" by simp
+ hence "x' \<in> H" ..
thus False by contradiction
qed
thus ?thesis by (simp!)
qed
qed
- show "graph H' h' \\<in> M"
- txt {* and $h'$ is norm-preserving. \skp *}
+ show "graph H' h' \<in> M"
+ -- {* and $h'$ is norm-preserving. \skp *}
proof -
- have "graph H' h' \\<in> norm_pres_extensions E p F f"
+ have "graph H' h' \<in> norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
show "is_linearform H' h'"
- by (rule h0_lf [OF _ _ _ _ _ _ x']) (simp!)+
- show "is_subspace H' E"
+ by (rule h'_lf) (simp! add: x')+
+ show "is_subspace H' E"
by (unfold H'_def) (rule vs_sum_subspace [OF _ lin_subspace])
have "is_subspace F H" .
also from h lin_vs
@@ -201,49 +260,259 @@
finally (subspace_trans [OF _ h])
show f_h': "is_subspace F H'" .
- show "graph F f \\<subseteq> graph H' h'"
+ show "graph F f \<subseteq> graph H' h'"
proof (rule graph_extI)
- fix x assume "x \\<in> F"
+ fix x assume "x \<in> F"
have "f x = h x" ..
also have " ... = h x + #0 * xi" by simp
also have "... = (let (y,a) = (x, #0) in h y + a * xi)"
by (simp add: Let_def)
also have
- "(x, #0) = (SOME (y, a). x = y + a (*) x' & y \\<in> H)"
- by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x']) (simp!)+
+ "(x, #0) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"
+ proof (rule decomp_H'_H [RS sym]) qed (simp! add: x')+
also have
- "(let (y,a) = (SOME (y,a). x = y + a (*) x' & y \\<in> H)
+ "(let (y,a) = (SOME (y,a). x = y + a \<cdot> x' \<and> y \<in> H)
in h y + a * xi)
= h' x" by (simp!)
finally show "f x = h' x" .
next
- from f_h' show "F \\<subseteq> H'" ..
+ from f_h' show "F \<subseteq> H'" ..
qed
- show "\\<forall>x \\<in> H'. h' x \\<le> p x"
- by (rule h0_norm_pres [OF _ _ _ _ x'])
+ show "\<forall>x \<in> H'. h' x \<le> p x"
+ by (rule h'_norm_pres) (assumption+, rule x')
qed
- thus "graph H' h' \\<in> M" by (simp!)
+ thus "graph H' h' \<in> M" by (simp!)
qed
qed
qed
qed
- hence "\\<not>(\\<forall>x \\<in> M. g \\<subseteq> x \\<longrightarrow> g = x)" by simp
+ hence "\<not> (\<forall>x \<in> M. g \<subseteq> x --> g = x)" by simp
-- {* So the graph $g$ of $h$ cannot be maximal. Contradiction! \skp *}
thus "H = E" by contradiction
qed
- thus "\\<exists>h. is_linearform E h \\<and> (\\<forall>x \\<in> F. h x = f x)
- \\<and> (\\<forall>x \\<in> E. h x \\<le> p x)"
+ thus "\<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x)
+ \<and> (\<forall>x \<in> E. h x \<le> p x)"
proof (intro exI conjI)
assume eq: "H = E"
from eq show "is_linearform E h" by (simp!)
- show "\\<forall>x \\<in> F. h x = f x"
+ show "\<forall>x \<in> F. h x = f x"
proof (intro ballI, rule sym)
- fix x assume "x \\<in> F" show "f x = h x " ..
+ fix x assume "x \<in> F" show "f x = h x " ..
qed
- from eq show "\\<forall>x \\<in> E. h x \\<le> p x" by (force!)
+ from eq show "\<forall>x \<in> E. h x \<le> p x" by (force!)
qed
qed
qed
-qed text_raw {* \smallskip\\ *}
+qed
+
+
+
+subsection {* Alternative formulation *}
+
+text {* The following alternative formulation of the Hahn-Banach
+Theorem\label{abs-HahnBanach} uses the fact that for a real linear form
+$f$ and a seminorm $p$ the
+following inequations are equivalent:\footnote{This was shown in lemma
+$\idt{abs{\dsh}ineq{\dsh}iff}$ (see page \pageref{abs-ineq-iff}).}
+\begin{matharray}{ll}
+\forall x\in H.\ap |h\ap x|\leq p\ap x& {\rm and}\\
+\forall x\in H.\ap h\ap x\leq p\ap x\\
+\end{matharray}
+*}
+
+theorem abs_HahnBanach:
+ "[| is_vectorspace E; is_subspace F E; is_linearform F f;
+ is_seminorm E p; \<forall>x \<in> F. |f x| <= p x |]
+ ==> \<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> (\<forall>x \<in> E. |g x| <= p x)"
+proof -
+ assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
+ "is_linearform F f" "\<forall>x \<in> F. |f x| <= p x"
+ have "\<forall>x \<in> F. f x <= p x" by (rule abs_ineq_iff [RS iffD1])
+ hence "\<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> (\<forall>x \<in> E. g x <= p x)"
+ by (simp! only: HahnBanach)
+ thus ?thesis
+ proof (elim exE conjE)
+ fix g assume "is_linearform E g" "\<forall>x \<in> F. g x = f x"
+ "\<forall>x \<in> E. g x <= p x"
+ hence "\<forall>x \<in> E. |g x| <= p x"
+ by (simp! add: abs_ineq_iff [OF subspace_refl])
+ thus ?thesis by (intro exI conjI)
+ qed
+qed
+
+subsection {* The Hahn-Banach Theorem for normed spaces *}
+
+text {* Every continuous linear form $f$ on a subspace $F$ of a
+norm space $E$, can be extended to a continuous linear form $g$ on
+$E$ such that $\fnorm{f} = \fnorm {g}$. *}
+
+theorem norm_HahnBanach:
+ "[| is_normed_vectorspace E norm; is_subspace F E;
+ is_linearform F f; is_continuous F norm f |]
+ ==> \<exists>g. is_linearform E g
+ \<and> is_continuous E norm g
+ \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> \<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
+proof -
+ assume e_norm: "is_normed_vectorspace E norm"
+ assume f: "is_subspace F E" "is_linearform F f"
+ assume f_cont: "is_continuous F norm f"
+ have e: "is_vectorspace E" ..
+ hence f_norm: "is_normed_vectorspace F norm" ..
+
+ txt{* We define a function $p$ on $E$ as follows:
+ \begin{matharray}{l}
+ p \: x = \fnorm f \cdot \norm x\\
+ \end{matharray}
+ *}
+
+ def p == "\<lambda>x. \<parallel>f\<parallel>F,norm * norm x"
+
+ txt{* $p$ is a seminorm on $E$: *}
+
+ have q: "is_seminorm E p"
+ proof
+ fix x y a assume "x \<in> E" "y \<in> E"
+
+ txt{* $p$ is positive definite: *}
+
+ show "#0 <= p x"
+ proof (unfold p_def, rule real_le_mult_order1a) thm fnorm_ge_zero
+ from f_cont f_norm show "#0 <= \<parallel>f\<parallel>F,norm" ..
+ show "#0 <= norm x" ..
+ qed
+
+ txt{* $p$ is absolutely homogenous: *}
+
+ show "p (a \<cdot> x) = |a| * p x"
+ proof -
+ have "p (a \<cdot> x) = \<parallel>f\<parallel>F,norm * norm (a \<cdot> x)"
+ by (simp!)
+ also have "norm (a \<cdot> x) = |a| * norm x"
+ by (rule normed_vs_norm_abs_homogenous)
+ also have "\<parallel>f\<parallel>F,norm * ( |a| * norm x )
+ = |a| * (\<parallel>f\<parallel>F,norm * norm x)"
+ by (simp! only: real_mult_left_commute)
+ also have "... = |a| * p x" by (simp!)
+ finally show ?thesis .
+ qed
+
+ txt{* Furthermore, $p$ is subadditive: *}
+
+ show "p (x + y) <= p x + p y"
+ proof -
+ have "p (x + y) = \<parallel>f\<parallel>F,norm * norm (x + y)"
+ by (simp!)
+ also
+ have "... <= \<parallel>f\<parallel>F,norm * (norm x + norm y)"
+ proof (rule real_mult_le_le_mono1a)
+ from f_cont f_norm show "#0 <= \<parallel>f\<parallel>F,norm" ..
+ show "norm (x + y) <= norm x + norm y" ..
+ qed
+ also have "... = \<parallel>f\<parallel>F,norm * norm x
+ + \<parallel>f\<parallel>F,norm * norm y"
+ by (simp! only: real_add_mult_distrib2)
+ finally show ?thesis by (simp!)
+ qed
+ qed
+
+ txt{* $f$ is bounded by $p$. *}
+
+ have "\<forall>x \<in> F. |f x| <= p x"
+ proof
+ fix x assume "x \<in> F"
+ from f_norm show "|f x| <= p x"
+ by (simp! add: norm_fx_le_norm_f_norm_x)
+ qed
+
+ txt{* Using the fact that $p$ is a seminorm and
+ $f$ is bounded by $p$ we can apply the Hahn-Banach Theorem
+ for real vector spaces.
+ So $f$ can be extended in a norm-preserving way to some function
+ $g$ on the whole vector space $E$. *}
+
+ with e f q
+ have "\<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> (\<forall>x \<in> E. |g x| <= p x)"
+ by (simp! add: abs_HahnBanach)
+
+ thus ?thesis
+ proof (elim exE conjE)
+ fix g
+ assume "is_linearform E g" and a: "\<forall>x \<in> F. g x = f x"
+ and b: "\<forall>x \<in> E. |g x| <= p x"
+
+ show "\<exists>g. is_linearform E g
+ \<and> is_continuous E norm g
+ \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> \<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
+ proof (intro exI conjI)
+
+ txt{* We furthermore have to show that
+ $g$ is also continuous: *}
+
+ show g_cont: "is_continuous E norm g"
+ proof
+ fix x assume "x \<in> E"
+ with b show "|g x| <= \<parallel>f\<parallel>F,norm * norm x"
+ by (simp add: p_def)
+ qed
+
+ txt {* To complete the proof, we show that
+ $\fnorm g = \fnorm f$. \label{order_antisym} *}
+
+ show "\<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
+ (is "?L = ?R")
+ proof (rule order_antisym)
+
+ txt{* First we show $\fnorm g \leq \fnorm f$. The function norm
+ $\fnorm g$ is defined as the smallest $c\in\bbbR$ such that
+ \begin{matharray}{l}
+ \All {x\in E} {|g\ap x| \leq c \cdot \norm x}
+ \end{matharray}
+ Furthermore holds
+ \begin{matharray}{l}
+ \All {x\in E} {|g\ap x| \leq \fnorm f \cdot \norm x}
+ \end{matharray}
+ *}
+
+ have "\<forall>x \<in> E. |g x| <= \<parallel>f\<parallel>F,norm * norm x"
+ proof
+ fix x assume "x \<in> E"
+ show "|g x| <= \<parallel>f\<parallel>F,norm * norm x"
+ by (simp!)
+ qed
+
+ with g_cont e_norm show "?L <= ?R"
+ proof (rule fnorm_le_ub)
+ from f_cont f_norm show "#0 <= \<parallel>f\<parallel>F,norm" ..
+ qed
+
+ txt{* The other direction is achieved by a similar
+ argument. *}
+
+ have "\<forall>x \<in> F. |f x| <= \<parallel>g\<parallel>E,norm * norm x"
+ proof
+ fix x assume "x \<in> F"
+ from a have "g x = f x" ..
+ hence "|f x| = |g x|" by simp
+ also from g_cont
+ have "... <= \<parallel>g\<parallel>E,norm * norm x"
+ proof (rule norm_fx_le_norm_f_norm_x)
+ show "x \<in> E" ..
+ qed
+ finally show "|f x| <= \<parallel>g\<parallel>E,norm * norm x" .
+ qed
+ thus "?R <= ?L"
+ proof (rule fnorm_le_ub [OF f_cont f_norm])
+ from g_cont show "#0 <= \<parallel>g\<parallel>E,norm" ..
+ qed
+ qed
+ qed
+ qed
+qed
+
end
\ No newline at end of file