theory HahnBanach = HahnBanachLemmas: text_raw {* \smallskip\\ *} (* from ~/Pub/TYPES99/HB/HahnBanach.thy *)
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 $E$ 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"
-- {* 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 $\Union c \in M$. *}
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)"
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" thm sup_subE [OF _ _ _ a]
by (rule sup_subE [OF _ _ _ a]) (simp !)+
(* FIXME 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!)+ *)
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 *}
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"
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 $g \in M$. \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 *}
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)
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"
txt {* 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> \<zero>"
proof (rule classical)
presume "x' = \<zero>"
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 *}
proof -
from h have "EX xi. ALL y:H. - p (y + x') - h y <= xi
& xi <= p (y + x') - h y"
proof (rule ex_xi)
fix u v assume "u:H" "v: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) = \\<epsilon>(y,a). x = y + a \<prod> 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'"
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 \<prod> x' & y \\<in> H)
= (t, #0)"
by (rule decomp_H0_H [OF _ _ _ _ _ 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' = \<zero> + x'" by (simp!)
from h show "\<zero> \\<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"
txt {* 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 h0_lf [OF _ _ _ _ _ _ x']) (simp!)+
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 (*) x' & y \\<in> H)"
by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x']) (simp!)+
also have
"(let (y,a) = (SOME (y,a). x = y + a (*) x' & 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 h0_norm_pres [OF _ _ _ _ x'])
qed
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
-- {* 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)"
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 \\<le> p x" by (force!)
qed
qed
qed
qed text_raw {* \smallskip\\ *}
end