(* Title: HOL/Hahn_Banach/Hahn_Banach.thy
Author: Gertrud Bauer, TU Munich
*)
header {* The Hahn-Banach Theorem *}
theory Hahn_Banach
imports Hahn_Banach_Lemmas
begin
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 Hahn_Banach:
assumes E: "vectorspace E" and "subspace F E"
and "seminorm E p" and "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 -
interpret vectorspace E by fact
interpret subspace F E by fact
interpret seminorm E p by fact
interpret linearform F f by fact
def M \<equiv> "norm_pres_extensions E p F f"
then have M: "M = \<dots>" by (simp only:)
from E have F: "vectorspace F" ..
note FE = `F \<unlhd> E`
{
fix c assume cM: "c \<in> chains 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"}. *}
unfolding M_def
proof (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 FE E have "?H \<unlhd> E"
by (rule sup_subE)
moreover from a M cM ex FE 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
}
then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"
-- {* 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"
unfolding M_def
proof (rule norm_pres_extensionI2)
show "linearform F f" by fact
show "F \<unlhd> E" by fact
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" by fact
qed
qed
then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
by blast
from gM 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" unfolding M_def ..
-- {* @{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 E have H: "vectorspace H"
by (rule subspace.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)
with `x' \<notin> H` 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)
from x'E have "vectorspace (lin x')" ..
with H show "H \<unlhd> H + lin x'" ..
qed
obtain xi where
xi: "\<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: 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 E 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 by (blast intro: that)
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"
from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
using `x' \<notin> H` `x' \<in> E` `x' \<noteq> 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'"
unfolding H'_def
proof
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
then have "(x', h' x') \<in> graph H' h'" ..
with eq have "(x', h' x') \<in> graph H h" by (simp only:)
then have "x' \<in> H" ..
with `x' \<notin> H` show 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'"
using h'_def H'_def HE linearform `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E
by (rule h'_lf)
show "H' \<unlhd> E"
unfolding H'_def
proof
show "H \<unlhd> E" by fact
show "vectorspace E" by fact
from x'E show "lin x' \<unlhd> E" ..
qed
from H `F \<unlhd> H` 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)"
using E HE
proof (rule decomp_H'_H [symmetric])
from FH x show "x \<in> H" ..
from x' show "x' \<noteq> 0" .
show "x' \<notin> H" by fact
show "x' \<in> E" by fact
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"
using h'_def H'_def `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E HE
`seminorm E p` linearform and hp xi
by (rule h'_norm_pres)
qed
qed
ultimately show ?thesis ..
qed
then have "\<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 *}
with gx 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-Hahn-Banach} 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_Hahn_Banach:
assumes E: "vectorspace E" and FE: "subspace F E"
and lf: "linearform F f" and sn: "seminorm 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 -
interpret vectorspace E by fact
interpret subspace F E by fact
interpret linearform F f by fact
interpret seminorm E p by fact
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)"
using E FE sn lf
proof (rule Hahn_Banach)
show "\<forall>x \<in> F. f x \<le> p x"
using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])
qed
then obtain g where lg: "linearform E g" and *: "\<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"
using _ E sn lg **
proof (rule abs_ineq_iff [THEN iffD2])
show "E \<unlhd> E" ..
qed
with lg * 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_Hahn_Banach:
fixes V and norm ("\<parallel>_\<parallel>")
fixes B defines "\<And>V f. B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"
fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"
and linearform: "linearform F f" and "continuous F f norm"
shows "\<exists>g. linearform E g
\<and> continuous E g norm
\<and> (\<forall>x \<in> F. g x = f x)
\<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
proof -
interpret normed_vectorspace E norm by fact
interpret normed_vectorspace_with_fn_norm E norm B fn_norm
by (auto simp: B_def fn_norm_def) intro_locales
interpret subspace F E by fact
interpret linearform F f by fact
interpret continuous F f norm by fact
have E: "vectorspace E" by intro_locales
have F: "vectorspace F" by rule intro_locales
have F_norm: "normed_vectorspace F norm"
using FE E_norm by (rule subspace_normed_vs)
have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm `continuous F f norm` , folded B_def fn_norm_def])
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: *}
have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
moreover from x have "0 \<le> \<parallel>x\<parallel>" ..
ultimately show "0 \<le> p x"
by (simp add: p_def zero_le_mult_iff)
txt {* @{text p} is absolutely homogeneous: *}
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 a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
by (simp add: mult_left_mono)
also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left)
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"
with `continuous F f norm` and linearform
show "\<bar>f x\<bar> \<le> p x"
unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm, folded B_def fn_norm_def])
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_Hahn_Banach [elim_format]) iprover
txt {* We furthermore have to show that @{text g} is also continuous: *}
have g_cont: "continuous E g norm" 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}
*}
from g_cont _ ge_zero
show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
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 {* The other direction is achieved by a similar argument. *}
show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
proof (rule normed_vectorspace_with_fn_norm.fn_norm_least
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm, folded B_def fn_norm_def])
fix x assume x: "x \<in> F"
show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
proof -
from a x have "g x = f x" ..
then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
also from g_cont
have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])
from FE x show "x \<in> E" ..
qed
finally show ?thesis .
qed
next
show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
using g_cont
by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
show "continuous F f norm" by fact
qed
qed
with linearformE a g_cont show ?thesis by blast
qed
end