(* 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; is_subspace F E; is_seminorm E p;
is_linearform F f; \\<forall>x \\<in> F. f x <= p x |]
==> \\<exists>h. is_linearform E h \\<and> (\\<forall>x \\<in> F. h x = f x)
\\<and> (\\<forall>x \\<in> E. h x <= 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 <= 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"
-- {* 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
\\<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 <= 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 <= p x"
by (rule sup_norm_pres, rule a) (simp!)+
qed
qed
}
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)
-- {* 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!)
qed
thus ?thesis
proof
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 <= 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 \\<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 <= 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 $h$ 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"
-- {* Pick $x' \in E \setminus H$. \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
thus ?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' == "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 <= xi
\\<and> xi <= 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 "\\<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 \\<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)"
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 ... <= p (v + x') + p (u + x')"
by (rule seminorm_diff_subadditive) (simp!)+
finally have "h v - h u <= p (v + x') + p (u + x')" .
thus "- p (u + x') - h u <= p (v + x') - h v"
by (rule real_diff_ineq_swap)
qed
thus ?thesis by rule rule
qed
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'"
-- {* 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'"
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 $h'$ 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 [fold 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 [RS sym]) (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 <= p x"
by (rule h'_norm_pres) (assumption+, rule x')
qed
thus "graph H' h' \\<in> M" by (simp!)
qed
qed
qed
qed
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 <= 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 (intro ballI, rule sym)
fix x assume "x \\<in> F" show "f x = h x " ..
qed
from eq show "\\<forall>x \\<in> E. h x <= p x" by (force!)
qed
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
$f$ 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)
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