(* 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 {*
\textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real
vector space @{text E}, let @{text p} be a semi-norm on @{text E},
and @{text f} be a linear form defined on @{text F} such that @{text
f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in> F. f x \<le> p x"}. Then
@{text f} can be extended to a linear form @{text h} on @{text E}
such that @{text h} is norm-preserving, i.e. @{text h} is also
bounded by @{text p}.
\bigskip
\textbf{Proof Sketch.}
\begin{enumerate}
\item Define @{text M} as the set of norm-preserving extensions of
@{text f} to subspaces of @{text E}. The linear forms in @{text M}
are ordered by domain extension.
\item We show that every non-empty chain in @{text M} has an upper
bound in @{text M}.
\item With Zorn's Lemma we conclude that there is a maximal function
@{text g} in @{text M}.
\item The domain @{text H} of @{text g} is the whole space @{text
E}, as shown by classical contradiction:
\begin{itemize}
\item Assuming @{text g} is not defined on whole @{text E}, it can
still be extended in a norm-preserving way to a super-space @{text
H'} of @{text H}.
\item Thus @{text g} can not be maximal. Contradiction!
\end{itemize}
\end{enumerate}
*}
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)"
-- {* Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E}, *}
-- {* and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p}, *}
-- {* then @{text f} can be extended to a linear form @{text h} on @{text 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"
-- {* Assume the context of the theorem. \skp *}
def M \<equiv> "norm_pres_extensions E p F f"
-- {* Define @{text M} as the set of all norm-preserving extensions of @{text F}. \skp *}
{
fix c assume "c \<in> chain M" "\<exists>x. x \<in> c"
have "\<Union>c \<in> M"
-- {* Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}: *}
-- {* @{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}. *}
proof (unfold M_def, rule norm_pres_extensionI)
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)"
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"
show "z = y" by (rule sup_definite)
qed
show "is_linearform ?H ?h"
by (simp! add: sup_lf a)
show "is_subspace ?H E"
by (rule sup_subE, rule a) (simp!)+
show "is_subspace F ?H"
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"
-- {* With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp *}
proof (rule Zorn's_Lemma)
-- {* We show that @{text 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!)
qed
thus ?thesis
proof
fix g assume "g \<in> M" "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
-- {* We consider such a maximal element @{text "g \<in> M"}. \skp *}
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"
-- {* @{text g} is a norm-preserving extension of @{text f}, in other words: *}
-- {* @{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E}, *}
-- {* and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp *}
proof -
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)
with that show ?thesis by blast
qed
have h: "is_vectorspace H" ..
have "H = E"
-- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}
proof (rule classical)
assume "H \<noteq> E"
-- {* Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended *}
-- {* in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp *}
have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
proof -
obtain x' where "x' \<in> E" "x' \<notin> H"
-- {* Pick @{text "x' \<in> E - H"}. \skp *}
proof -
have "\<exists>x' \<in> E. x' \<notin> H"
proof (rule set_less_imp_diff_not_empty)
have "H \<subseteq> E" ..
thus "H \<subset> E" ..
qed
with that show ?thesis by blast
qed
have x': "x' \<noteq> 0"
proof (rule classical)
presume "x' = 0"
with h have "x' \<in> H" by simp
thus ?thesis by contradiction
qed blast
def H' \<equiv> "H + lin x'"
-- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \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 @{text \<xi>} that fulfills certain inequations; this will *}
-- {* be used to establish that @{text h'} is a norm-preserving extension of @{text h}.
\label{ex-xi-use}\skp *}
proof -
from h have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
\<and> xi \<le> p (y + x') - h y"
proof (rule ex_xi)
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 "... \<le> p (v - u)"
by (simp!)
also have "v - u = x' + - x' + v + - u"
by (simp! add: diff_eq1)
also have "... = v + x' + - (u + x')"
by (simp!)
also have "... = (v + x') - (u + x')"
by (simp! add: diff_eq1)
also have "p ... \<le> p (v + x') + p (u + x')"
by (rule seminorm_diff_subadditive) (simp_all!)
finally have "h v - h u \<le> p (v + x') + p (u + x')" .
thus "- p (u + x') - h u \<le> p (v + x') - h v"
by (rule real_diff_ineq_swap)
qed
thus ?thesis ..
qed
def h' \<equiv> "\<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 @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp *}
show ?thesis
proof
show "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
-- {* Show that @{text h'} is an extension of @{text h} \dots \skp *}
proof
show "g \<subseteq> graph H' h'"
proof -
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 \<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'"
proof (rule subspace_subset)
show "is_subspace H H'"
proof (unfold H'_def, rule subspace_vs_sum1)
show "is_vectorspace H" ..
show "is_vectorspace (lin x')" ..
qed
qed
qed
thus ?thesis by (simp!)
qed
show "g \<noteq> graph H' h'"
proof -
have "graph H h \<noteq> graph H' h'"
proof
assume e: "graph H h = graph H' h'"
have "x' \<in> H'"
proof (unfold H'_def, rule vs_sumI)
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" ..
thus False by contradiction
qed
thus ?thesis by (simp!)
qed
qed
show "graph H' h' \<in> M"
-- {* and @{text h'} is norm-preserving. \skp *}
proof -
have "graph H' h' \<in> norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
show "is_linearform H' h'"
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
have [folded H'_def]: "is_subspace H (H + lin x')" ..
finally (subspace_trans [OF _ h])
show f_h': "is_subspace F H'" .
show "graph F f \<subseteq> graph H' h'"
proof (rule graph_extI)
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 \<cdot> x' \<and> y \<in> H)"
by (rule decomp_H'_H [symmetric]) (simp! add: x')+
also have
"(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'" ..
qed
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!)
qed
qed
qed
hence "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
-- {* So the graph @{text g} of @{text 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)"
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"
proof
fix x assume "x \<in> F" have "f x = h x " ..
thus "h x = f x" ..
qed
from eq show "\<forall>x \<in> E. h x \<le> p x" by (blast!)
qed
qed
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 @{text f} and a seminorm @{text p} the following inequations
are equivalent:\footnote{This was shown in lemma @{thm [source]
abs_ineq_iff} (see page \pageref{abs-ineq-iff}).}
\begin{center}
\begin{tabular}{lll}
@{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
@{text "\<forall>x \<in> H. h x \<le> p x"} \\
\end{tabular}
\end{center}
*}
theorem abs_HahnBanach:
"is_vectorspace E \<Longrightarrow> is_subspace F E \<Longrightarrow> is_linearform F f
\<Longrightarrow> is_seminorm E p \<Longrightarrow> \<forall>x \<in> F. \<bar>f x\<bar> \<le> p x
\<Longrightarrow> \<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
\<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
proof -
assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
"is_linearform F f" "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
have "\<forall>x \<in> F. f x \<le> p x" by (rule abs_ineq_iff [THEN iffD1])
hence "\<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
\<and> (\<forall>x \<in> E. g x \<le> 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 \<le> p x"
hence "\<forall>x \<in> E. \<bar>g x\<bar> \<le> 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 @{text f} on a subspace @{text F} of a
norm space @{text E}, can be extended to a continuous linear form
@{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}.
*}
theorem norm_HahnBanach:
"is_normed_vectorspace E norm \<Longrightarrow> is_subspace F E
\<Longrightarrow> is_linearform F f \<Longrightarrow> is_continuous F norm f
\<Longrightarrow> \<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 @{text p} on @{text E} as follows:
@{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
*}
def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>F,norm * norm x"
txt {* @{text p} is a seminorm on @{text E}: *}
have q: "is_seminorm E p"
proof
fix x y a assume "x \<in> E" "y \<in> E"
txt {* @{text p} is positive definite: *}
show "#0 \<le> p x"
proof (unfold p_def, rule real_le_mult_order1a)
from f_cont f_norm show "#0 \<le> \<parallel>f\<parallel>F,norm" ..
show "#0 \<le> norm x" ..
qed
txt {* @{text p} is absolutely homogenous: *}
show "p (a \<cdot> x) = \<bar>a\<bar> * 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) = \<bar>a\<bar> * norm x"
by (rule normed_vs_norm_abs_homogenous)
also have "\<parallel>f\<parallel>F,norm * (\<bar>a\<bar> * norm x )
= \<bar>a\<bar> * (\<parallel>f\<parallel>F,norm * norm x)"
by (simp! only: real_mult_left_commute)
also have "... = \<bar>a\<bar> * p x" by (simp!)
finally show ?thesis .
qed
txt {* Furthermore, @{text p} is subadditive: *}
show "p (x + y) \<le> p x + p y"
proof -
have "p (x + y) = \<parallel>f\<parallel>F,norm * norm (x + y)"
by (simp!)
also
have "... \<le> \<parallel>f\<parallel>F,norm * (norm x + norm y)"
proof (rule real_mult_le_le_mono1a)
from f_cont f_norm show "#0 \<le> \<parallel>f\<parallel>F,norm" ..
show "norm (x + y) \<le> 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 {* @{text f} is bounded by @{text p}. *}
have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
proof
fix x assume "x \<in> F"
from f_norm show "\<bar>f x\<bar> \<le> p x"
by (simp! add: norm_fx_le_norm_f_norm_x)
qed
txt {*
Using the fact that @{text p} is a seminorm and @{text f} is bounded
by @{text p} we can apply the Hahn-Banach Theorem for real vector
spaces. So @{text f} can be extended in a norm-preserving way to
some function @{text g} on the whole vector space @{text 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. \<bar>g x\<bar> \<le> 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. \<bar>g x\<bar> \<le> 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 @{text g} is also continuous:
*}
show g_cont: "is_continuous E norm g"
proof
fix x assume "x \<in> E"
with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
by (simp add: p_def)
qed
txt {*
To complete the proof, we show that
@{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. \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 @{text "\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>"}. The function norm @{text
"\<parallel>g\<parallel>"} is defined as the smallest @{text "c \<in> \<real>"} such that
\begin{center}
\begin{tabular}{l}
@{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{tabular}
\end{center}
\noindent Furthermore holds
\begin{center}
\begin{tabular}{l}
@{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
\end{tabular}
\end{center}
*}
have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
proof
fix x assume "x \<in> E"
show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
by (simp!)
qed
with g_cont e_norm show "?L \<le> ?R"
proof (rule fnorm_le_ub)
from f_cont f_norm show "#0 \<le> \<parallel>f\<parallel>F,norm" ..
qed
txt{* The other direction is achieved by a similar
argument. *}
have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x"
proof
fix x assume "x \<in> F"
from a have "g x = f x" ..
hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by simp
also from g_cont
have "... \<le> \<parallel>g\<parallel>E,norm * norm x"
proof (rule norm_fx_le_norm_f_norm_x)
show "x \<in> E" ..
qed
finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x" .
qed
thus "?R \<le> ?L"
proof (rule fnorm_le_ub [OF f_cont f_norm])
from g_cont show "#0 \<le> \<parallel>g\<parallel>E,norm" ..
qed
qed
qed
qed
qed
end