(* 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:
includes vectorspace E + subvectorspace F E +
seminorm_vectorspace E p + linearform F f
assumes fp: "\<forall>x \<in> F. f x \<le> p x"
shows "\<exists>h. 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 -
def M \<equiv> "norm_pres_extensions E p F f"
hence M: "M = \<dots>" by (simp only:)
have F: "vectorspace F" ..
{
fix c assume cM: "c \<in> chain M" and ex: "\<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)
let ?H = "domain (\<Union>c)"
let ?h = "funct (\<Union>c)"
have a: "graph ?H ?h = \<Union>c"
proof (rule graph_domain_funct)
fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"
with M_def cM show "z = y" by (rule sup_definite)
qed
moreover from M cM a have "linearform ?H ?h"
by (rule sup_lf)
moreover from a M cM ex have "?H \<unlhd> E"
by (rule sup_subE)
moreover from a M cM ex have "F \<unlhd> ?H"
by (rule sup_supF)
moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
by (rule sup_ext)
moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"
by (rule sup_norm_pres)
ultimately show "\<exists>H h. \<Union>c = 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)" by blast
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: *}
show "graph F f \<in> M"
proof (unfold M_def, rule norm_pres_extensionI2)
show "linearform F f" .
show "F \<unlhd> E" .
from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)
show "graph F f \<subseteq> graph F f" ..
show "\<forall>x\<in>F. f x \<le> p x" .
qed
qed
then obtain g where gM: "g \<in> M" and "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
by blast
from gM [unfolded M_def] obtain H h where
g_rep: "g = graph H h"
and linearform: "linearform H h"
and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"
and graphs: "graph F f \<subseteq> graph H h"
and hp: "\<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 *}
from HE have H: "vectorspace H"
by (rule subvectorspace.vectorspace)
have HE_eq: "H = E"
-- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}
proof (rule classical)
assume neq: "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 -
from HE have "H \<subseteq> E" ..
with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast
obtain x': "x' \<noteq> 0"
proof
show "x' \<noteq> 0"
proof
assume "x' = 0"
with H have "x' \<in> H" by (simp only: vectorspace.zero)
then show False by contradiction
qed
qed
def H' \<equiv> "H + lin x'"
-- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}
have HH': "H \<unlhd> H'"
proof (unfold H'_def)
have "vectorspace (lin x')" ..
with H show "H \<unlhd> H + lin x'" ..
qed
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: "u \<in> H" and v: "v \<in> H"
with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto
from H u v linearform have "h v - h u = h (v - u)"
by (simp add: vectorspace_linearform.diff)
also from hp and H u v have "\<dots> \<le> p (v - u)"
by (simp only: vectorspace.diff_closed)
also from x'E uE vE have "v - u = x' + - x' + v + - u"
by (simp add: diff_eq1)
also from x'E uE vE have "\<dots> = v + x' + - (u + x')"
by (simp add: add_ac)
also from x'E uE vE have "\<dots> = (v + x') - (u + x')"
by (simp add: diff_eq1)
also from x'E uE vE have "p \<dots> \<le> p (v + x') + p (u + x')"
by (simp add: diff_subadditive)
finally have "h v - h u \<le> p (v + x') + p (u + x')" .
then show "- p (u + x') - h u \<le> p (v + x') - h v"
by simp
qed
then show ?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 *}
have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
-- {* @{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: "t \<in> H"
have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
by (rule decomp_H'_H)
with h'_def show "h t = h' t" by (simp add: Let_def)
next
from HH' show "H \<subseteq> H'" ..
qed
with g_rep show ?thesis by (simp only:)
qed
show "g \<noteq> graph H' h'"
proof -
have "graph H h \<noteq> graph H' h'"
proof
assume eq: "graph H h = graph H' h'"
have "x' \<in> H'"
proof (unfold H'_def, rule)
from H show "0 \<in> H" by (rule vectorspace.zero)
from x'E show "x' \<in> lin x'" by (rule x_lin_x)
from x'E show "x' = 0 + x'" by simp
qed
hence "(x', h' x') \<in> graph H' h'" ..
with eq have "(x', h' x') \<in> graph H h" by (simp only:)
hence "x' \<in> H" ..
thus False by contradiction
qed
with g_rep show ?thesis by simp
qed
qed
moreover have "graph H' h' \<in> M"
-- {* and @{text h'} is norm-preserving. \skp *}
proof (unfold M_def)
show "graph H' h' \<in> norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
show "linearform H' h'" by (rule h'_lf)
show "H' \<unlhd> E"
proof (unfold H'_def, rule)
show "H \<unlhd> E" .
show "vectorspace E" .
from x'E show "lin x' \<unlhd> E" ..
qed
have "F \<unlhd> H" .
from H this HH' show FH': "F \<unlhd> H'"
by (rule vectorspace.subspace_trans)
show "graph F f \<subseteq> graph H' h'"
proof (rule graph_extI)
fix x assume x: "x \<in> F"
with graphs have "f x = h x" ..
also have "\<dots> = h x + 0 * xi" by simp
also have "\<dots> = (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)"
proof (rule decomp_H'_H [symmetric])
from FH x show "x \<in> H" ..
from x' show "x' \<noteq> 0" .
qed
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 only: h'_def)
finally show "f x = h' x" .
next
from FH' show "F \<subseteq> H'" ..
qed
show "\<forall>x \<in> H'. h' x \<le> p x" by (rule h'_norm_pres)
qed
qed
ultimately show ?thesis ..
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 *}
then show "H = E" by contradiction
qed
from HE_eq and linearform have "linearform E h"
by (simp only:)
moreover have "\<forall>x \<in> F. h x = f x"
proof
fix x assume "x \<in> F"
with graphs have "f x = h x" ..
then show "h x = f x" ..
qed
moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
by (simp only:)
ultimately show ?thesis by blast
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:
includes vectorspace E + subvectorspace F E +
linearform F f + seminorm_vectorspace E p
assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
shows "\<exists>g. 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 -
have "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x)
\<and> (\<forall>x \<in> E. g x \<le> p x)"
proof (rule HahnBanach)
show "\<forall>x \<in> F. f x \<le> p x"
by (rule abs_ineq_iff [THEN iffD1])
qed
then obtain g where * : "linearform E g" "\<forall>x \<in> F. g x = f x"
and "\<forall>x \<in> E. g x \<le> p x" by blast
have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
proof (rule abs_ineq_iff [THEN iffD2])
show "E \<unlhd> E" ..
qed
with * show ?thesis by blast
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:
includes functional_vectorspace E + subvectorspace F E +
linearform F f + continuous F norm f
shows "\<exists>g. linearform E g
\<and> continuous E norm g
\<and> (\<forall>x \<in> F. g x = f x)
\<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
proof -
have E: "vectorspace E" .
have E_norm: "normed_vectorspace E norm" ..
have FE: "F \<unlhd> E" .
have F: "vectorspace F" ..
have linearform: "linearform F f" .
have F_norm: "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>\<hyphen>F * \<parallel>x\<parallel>"
txt {* @{text p} is a seminorm on @{text E}: *}
have q: "seminorm E p"
proof
fix x y a assume x: "x \<in> E" and y: "y \<in> E"
txt {* @{text p} is positive definite: *}
show "0 \<le> p x"
proof (unfold p_def, rule real_le_mult_order1a)
show "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
apply (unfold function_norm_def B_def)
using normed_vectorspace.axioms [OF F_norm] ..
from x show "0 \<le> \<parallel>x\<parallel>" ..
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>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>"
by (simp only: p_def)
also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
by (rule abs_homogenous)
also have "\<parallel>f\<parallel>\<hyphen>F * (\<bar>a\<bar> * \<parallel>x\<parallel>) = \<bar>a\<bar> * (\<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>)"
by simp
also have "\<dots> = \<bar>a\<bar> * p x"
by (simp only: p_def)
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>\<hyphen>F * \<parallel>x + y\<parallel>"
by (simp only: p_def)
also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
proof (rule real_mult_le_le_mono1a)
show "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
apply (unfold function_norm_def B_def)
using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
from x y show "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
qed
also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>"
by (simp only: real_add_mult_distrib2)
also have "\<dots> = p x + p y"
by (simp only: p_def)
finally show ?thesis .
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"
show "\<bar>f x\<bar> \<le> p x"
apply (unfold p_def function_norm_def B_def)
using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
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 FE linearform q obtain g where
linearformE: "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"
by (rule abs_HahnBanach [elim_format]) rules
txt {* We furthermore have to show that @{text g} is also continuous: *}
have g_cont: "continuous E norm g" using linearformE
proof
fix x assume "x \<in> E"
with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
by (simp only: p_def)
qed
txt {* To complete the proof, we show that @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. *}
have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
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>\<hyphen>F * \<parallel>x\<parallel>"
proof
fix x assume "x \<in> E"
with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
by (simp only: p_def)
qed
show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
apply (unfold function_norm_def B_def)
apply rule
apply (rule normed_vectorspace.axioms [OF E_norm])+
apply (rule continuous.axioms [OF g_cont])+
apply (rule b [unfolded p_def function_norm_def B_def])
using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
txt {* The other direction is achieved by a similar argument. *}
have ** : "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
proof
fix x assume x: "x \<in> F"
from a have "g x = f x" ..
hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
also have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
apply (unfold function_norm_def B_def)
apply rule
apply (rule normed_vectorspace.axioms [OF E_norm])+
apply (rule continuous.axioms [OF g_cont])+
proof -
from FE x show "x \<in> E" ..
qed
finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
qed
show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
apply (unfold function_norm_def B_def)
apply rule
apply (rule normed_vectorspace.axioms [OF F_norm])+
apply assumption+
apply (rule ** [unfolded function_norm_def B_def])
apply rule
apply assumption+
apply (rule continuous.axioms [OF g_cont])+
done (* FIXME *)
qed
with linearformE a g_cont show ?thesis
by blast
qed
end