--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Mon Jan 03 14:07:10 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Mon Jan 03 17:33:34 2000 +0100
@@ -13,13 +13,298 @@
Theorem, closely following \cite[\S36]{Heuser:1986}.
*};
-subsection {* The case of general linear spaces *};
+subsection {* The Hahn-Banach Theorem for vector spaces *};
+
+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; ALL x:F. f x <= p x |]
+ ==> EX h. is_linearform E h & (ALL x:F. h x = f x)
+ & (ALL x:E. h x <= p x)";
+proof -;
+
+txt {* 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 $p$. *};
+
+ assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
+ "is_linearform F f" "ALL x:F. f x <= p x";
+
+txt {* Define $M$ as the set of all norm-preserving extensions of $F$. *};
+
+ def M == "norm_pres_extensions E p F f";
+ {{;
+ fix c; assume "c : chain M" "EX x. x:c";
+
+txt {* Show that every non-empty chain $c$ in $M$ has an upper bound in $M$: $\Union c$ is greater that every element of the chain $c$, so $\Union c$ is an upper bound of $c$ that lies in $M$. *};
+
+ have "Union c : M";
+ proof (unfold M_def, rule norm_pres_extensionI);
+ show "EX (H::'a set) h::'a => real. graph H h = Union c
+ & is_linearform H h
+ & is_subspace H E
+ & is_subspace F H
+ & graph F f <= graph H h
+ & (ALL x::'a: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) : Union c" "(x, z) : 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 <= graph ?H ?h";
+ by (rule sup_ext, rule a) (simp!)+;
+ show "ALL x::'a:?H. ?h x <= p x";
+ by (rule sup_norm_pres, rule a) (simp!)+;
+ qed;
+ qed;
+ }};
+
+txt {* With Zorn's Lemma we can conclude that there is a maximal element $g$ in $M$. *};
+
+ hence "EX g:M. ALL x:M. g <= x --> g = x";
+ proof (rule Zorn's_Lemma);
+ txt{* We show that $M$ is non-empty: *};
+ have "graph F f : 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 : M"; by (simp!);
+ qed;
+ thus ?thesis;
+ proof;
+
+txt {* We take this maximal element $g$. *};
+
+ fix g; assume "g:M" "ALL x:M. g <= x --> g = x";
+ show ?thesis;
+
+txt_raw {* \isamarkuptxt{$g$ is a norm-preserving extension of $f$, that is: $g$ is the graph of a linear form $h$, defined on a subspace $H$ of $E$, which is a superspace of $F$. $h$ is an extension of $f$ and $h$ is again bounded by $p$.} *};
+
+ obtain H h in "graph H h = g" and "is_linearform H h"
+ "is_subspace H E" "is_subspace F H" "graph F f <= graph H h"
+ "ALL x:H. h x <= p x";
+ proof -;
+ have "EX H h. graph H h = g & is_linearform H h
+ & is_subspace H E & is_subspace F H
+ & graph F f <= graph H h
+ & (ALL x:H. h x <= p x)"; by (simp! add: norm_pres_extension_D);
+ thus ?thesis; by (elim exE conjE) (rule that);
+ qed;
+ have h: "is_vectorspace H"; ..;
+
+txt {* We show that $h$ is defined on whole $E$ by classical contradiction. *};
+
+ have "H = E";
+ proof (rule classical);
+
+txt_raw {* \isamarkuptxt{Assume $h$ is not defined on whole $E$.} *};
+
+ assume "H ~= E";
+
+txt {* Then show that $h$ can be extended in a norm-preserving way to a function $h_0$ with the graph $g_{h0}$. *};
+
+ have "EX g_h0 : M. g <= g_h0 & g ~= g_h0";
+
+txt_raw {* \isamarkuptxt{Take $x_0 \in E \setminus H$.} *};
+
+ obtain x0 in "x0:E" "x0~:H";
+ proof -;
+ have "EX x0:E. x0~:H";
+ proof (rule set_less_imp_diff_not_empty);
+ have "H <= E"; ..;
+ thus "H < E"; ..;
+ qed;
+ thus ?thesis; by (elim bexE) (rule that);
+ qed;
+ have x0: "x0 ~= <0>";
+ proof (rule classical);
+ presume "x0 = <0>";
+ with h; have "x0:H"; by simp;
+ thus ?thesis; by contradiction;
+ qed blast;
+
+txt {* Define $H_0$ as the direct sum of $H$ and the linear closure of $x_0$. *};
+
+ def H0 == "H + lin x0";
+ show ?thesis;
+
+txt_raw {* \isamarkuptxt{Chose a real number $\xi$ that fulfills certain inequations, which will be used to establish that $h_0$ is a norm-preserving extension of $h$.} *};
-text {* Let $f$ be a linear form $f$ defined on a subspace $F$
- of a norm vector space $E$. If $f$ is
- bounded by some seminorm $q$ on $E$, it can be extended
- to a function on $E$ in a norm-preserving way. *};
+ obtain xi in "ALL y:H. - p (y + x0) - h y <= xi
+ & xi <= p (y + x0) - h y";
+ proof -;
+ from h; have "EX xi. ALL y:H. - p (y + x0) - h y <= xi
+ & xi <= p (y + x0) - 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 = x0 + - x0 + v + - u";
+ by (simp! add: diff_eq1);
+ also; have "... = v + x0 + - (u + x0)";
+ by (simp!);
+ also; have "... = (v + x0) - (u + x0)";
+ by (simp! add: diff_eq1);
+ also; have "p ... <= p (v + x0) + p (u + x0)";
+ by (rule seminorm_diff_subadditive) (simp!)+;
+ finally; have "h v - h u <= p (v + x0) + p (u + x0)"; .;
+
+ thus "- p (u + x0) - h u <= p (v + x0) - h v";
+ by (rule real_diff_ineq_swap);
+ qed;
+ thus ?thesis; by (elim exE) (rule that);
+ qed;
+
+txt {* Define the extension $h_0$ of $h$ to $H_0$ using $\xi$. *};
+ def h0 == "\<lambda>x. let (y,a) = SOME (y, a). x = y + a <*> x0
+ & y:H
+ in (h y) + a * xi";
+ show ?thesis;
+ proof;
+
+txt {* Show that $h_0$ is an extension of $h$ *};
+
+ show "g <= graph H0 h0 & g ~= graph H0 h0";
+ proof;
+ show "g <= graph H0 h0";
+ proof -;
+ have "graph H h <= graph H0 h0";
+ proof (rule graph_extI);
+ fix t; assume "t:H";
+ have "(SOME (y, a). t = y + a <*> x0 & y : H)
+ = (t,0r)";
+ by (rule decomp_H0_H, rule x0);
+ thus "h t = h0 t"; by (simp! add: Let_def);
+ next;
+ show "H <= H0";
+ proof (rule subspace_subset);
+ show "is_subspace H H0";
+ proof (unfold H0_def, rule subspace_vs_sum1);
+ show "is_vectorspace H"; ..;
+ show "is_vectorspace (lin x0)"; ..;
+ qed;
+ qed;
+ qed;
+ thus ?thesis; by (simp!);
+ qed;
+ show "g ~= graph H0 h0";
+ proof -;
+ have "graph H h ~= graph H0 h0";
+ proof;
+ assume e: "graph H h = graph H0 h0";
+ have "x0 : H0";
+ proof (unfold H0_def, rule vs_sumI);
+ show "x0 = <0> + x0"; by (simp!);
+ from h; show "<0> : H"; ..;
+ show "x0 : lin x0"; by (rule x_lin_x);
+ qed;
+ hence "(x0, h0 x0) : graph H0 h0"; ..;
+ with e; have "(x0, h0 x0) : graph H h"; by simp;
+ hence "x0 : H"; ..;
+ thus False; by contradiction;
+ qed;
+ thus ?thesis; by (simp!);
+ qed;
+ qed;
+
+txt {* and $h_0$ is norm-preserving. *};
+
+ show "graph H0 h0 : M";
+ proof -;
+ have "graph H0 h0 : norm_pres_extensions E p F f";
+ proof (rule norm_pres_extensionI2);
+ show "is_linearform H0 h0";
+ by (rule h0_lf, rule x0) (simp!)+;
+ show "is_subspace H0 E";
+ by (unfold H0_def, rule vs_sum_subspace,
+ rule lin_subspace);
+ have "is_subspace F H"; .;
+ also; from h lin_vs;
+ have [fold H0_def]: "is_subspace H (H + lin x0)"; ..;
+ finally (subspace_trans [OF _ h]);
+ show f_h0: "is_subspace F H0"; .;
+
+ show "graph F f <= graph H0 h0";
+ proof (rule graph_extI);
+ fix x; assume "x:F";
+ have "f x = h x"; ..;
+ also; have " ... = h x + 0r * xi"; by simp;
+ also; have "... = (let (y,a) = (x, 0r) in h y + a * xi)";
+ by (simp add: Let_def);
+ also; have
+ "(x, 0r) = (SOME (y, a). x = y + a <*> x0 & y : H)";
+ by (rule decomp_H0_H [RS sym], rule x0) (simp!)+;
+ also; have
+ "(let (y,a) = (SOME (y,a). x = y + a <*> x0 & y : H)
+ in h y + a * xi)
+ = h0 x"; by (simp!);
+ finally; show "f x = h0 x"; .;
+ next;
+ from f_h0; show "F <= H0"; ..;
+ qed;
+
+ show "ALL x:H0. h0 x <= p x";
+ by (rule h0_norm_pres, rule x0);
+ qed;
+ thus "graph H0 h0 : M"; by (simp!);
+ qed;
+ qed;
+ qed;
+ qed;
+
+txt {* So the graph $g$ of $h$ cannot be maximal. Contradiction. *};
+
+ hence "~ (ALL x:M. g <= x --> g = x)"; by simp;
+ thus ?thesis; by contradiction;
+ qed;
+
+txt {* Now we have a linear extension $h$ of $f$ to $E$ that is
+bounded by $p$. *};
+
+ thus "EX h. is_linearform E h & (ALL x:F. h x = f x)
+ & (ALL x:E. h x <= p x)";
+ proof (intro exI conjI);
+ assume eq: "H = E";
+ from eq; show "is_linearform E h"; by (simp!);
+ show "ALL x:F. h x = f x";
+ proof (intro ballI, rule sym);
+ fix x; assume "x:F"; show "f x = h x "; ..;
+ qed;
+ from eq; show "ALL x:E. h x <= p x"; by (force!);
+ qed;
+ qed;
+ qed;
+qed;
+(*
theorem HahnBanach:
"[| is_vectorspace E; is_subspace F E; is_seminorm E p;
is_linearform F f; ALL x:F. f x <= p x |]
@@ -43,14 +328,13 @@
thus "is_subspace F F"; ..;
qed (blast!)+;
- subsubsect {* Existence of a limit function the norm-preserving
- extensions *};
+ subsubsect {* Existence of a limit function *};
txt {* For every non-empty chain of norm-preserving extensions
the union of all functions in the chain is again a norm-preserving
extension. (The union is an upper bound for all elements.
It is even the least upper bound, because every upper bound of $M$
- is also an upper bound for $\Union c$.) *};
+ is also an upper bound for $\Union c$, as $\Union c\in M$) *};
{{;
fix c; assume "c:chain M" "EX x. x:c";
@@ -72,7 +356,6 @@
fix x y z; assume "(x, y) : Union c" "(x, z) : 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";
@@ -122,19 +405,19 @@
proof (rule classical);
txt {* Assume that the domain of the supremum is not $E$, *};
-;
+
assume "H ~= E";
have "H <= E"; ..;
hence "H < E"; ..;
- txt{* then there exists an element $x_0$ in $E \ H$: *};
+ txt{* then there exists an element $x_0$ in $E \setminus H$: *};
hence "EX x0:E. x0~:H";
by (rule set_less_imp_diff_not_empty);
txt {* We get that $h$ can be extended in a
norm-preserving way to some $H_0$. *};
-;
+
hence "EX H0 h0. g <= graph H0 h0 & g ~= graph H0 h0
& graph H0 h0 : M";
proof;
@@ -187,7 +470,7 @@
"\<lambda>x. let (y,a) = SOME (y, a). x = y + a <*> x0 & y:H
in (h y) + a * xi";
- txt {* We get that the graph of $h_0$ extend that of
+ txt {* We get that the graph of $h_0$ extends that of
$h$. *};
have "graph H h <= graph H0 h0";
@@ -270,7 +553,7 @@
qed;
show "ALL x:H0. h0 x <= p x";
- by (rule h0_norm_pres, rule x0) (assumption | (simp!))+;
+ by (rule h0_norm_pres, rule x0) (assumption | simp!)+;
qed;
thus "graph H0 h0 : M"; by (simp!);
qed;
@@ -303,13 +586,14 @@
qed;
qed;
qed;
-
+*)
-subsection {* An alternative formulation of the theorem *};
+subsection {* Alternative formulation *};
text {* The following alternative formulation of the Hahn-Banach
-Theorem\label{rabs-HahnBanach} uses the fact that for real numbers the
+Theorem\label{rabs-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
$\idt{rabs{\dsh}ineq{\dsh}iff}$ (see page \pageref{rabs-ineq-iff}).}
\begin{matharray}{ll}
@@ -319,32 +603,27 @@
*};
theorem rabs_HahnBanach:
- "[| is_vectorspace E; is_subspace F E; is_seminorm E p;
- is_linearform F f; ALL x:F. rabs (f x) <= p x |]
- ==> EX g. is_linearform E g
- & (ALL x:F. g x = f x)
- & (ALL x:E. rabs (g x) <= p x)";
-proof -;
- assume e: "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
- "is_linearform F f" "ALL x:F. rabs (f x) <= p x";
- have "ALL x:F. f x <= p x";
- by (rule rabs_ineq_iff [RS iffD1]);
+ "[| is_vectorspace E; is_subspace F E; is_linearform F f;
+ is_seminorm E p; ALL x:F. rabs (f x) <= p x |]
+ ==> EX g. is_linearform E g & (ALL x:F. g x = f x)
+ & (ALL x:E. rabs (g x) <= p x)";
+proof -;
+ assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
+ "is_linearform F f" "ALL x:F. rabs (f x) <= p x";
+ have "ALL x:F. f x <= p x"; by (rule rabs_ineq_iff [RS iffD1]);
hence "EX g. is_linearform E g & (ALL x:F. g x = f x)
& (ALL x:E. g x <= p x)";
by (simp! only: HahnBanach);
- thus ?thesis;
+ thus ?thesis;
proof (elim exE conjE);
fix g; assume "is_linearform E g" "ALL x:F. g x = f x"
"ALL x:E. g x <= p x";
- show ?thesis;
- proof (intro exI conjI);
- from e; show "ALL x:E. rabs (g x) <= p x";
- by (simp! add: rabs_ineq_iff [OF subspace_refl]);
- qed;
+ hence "ALL x:E. rabs (g x) <= p x";
+ by (simp! add: rabs_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
@@ -365,9 +644,9 @@
have e: "is_vectorspace E"; ..;
with _; have f_norm: "is_normed_vectorspace F norm"; ..;
- txt{* We define the function $p$ on $E$ as follows:
+ txt{* We define a function $p$ on $E$ as follows:
\begin{matharray}{l}
- p \ap x = \fnorm f \cdot \norm x\\
+ p \: x = \fnorm f \cdot \norm x\\
\end{matharray}
*};
@@ -431,7 +710,7 @@
qed;
txt{* Using the fact that $p$ is a seminorm and
- $f$ is bounded by $q$ we can apply the Hahn-Banach Theorem
+ $f$ 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 $E$. *};
@@ -453,7 +732,7 @@
& function_norm E norm g = function_norm F norm f";
proof (intro exI conjI);
- txt{* We futhermore have to show that
+ txt{* We furthermore have to show that
$g$ is also continuous: *};
show g_cont: "is_continuous E norm g";
@@ -465,7 +744,7 @@
qed;
txt {* To complete the proof, we show that
- $\fnorm g = \fnorm f$. *};
+ $\fnorm g = \fnorm f$. \label{order_antisym} *};
show "function_norm E norm g = function_norm F norm f"
(is "?L = ?R");
@@ -502,15 +781,15 @@
fix x; assume "x : F";
from a; have "g x = f x"; ..;
hence "rabs (f x) = rabs (g x)"; by simp;
- also; from _ _ g_cont;
+ also; from g_cont;
have "... <= function_norm E norm g * norm x";
- by (rule norm_fx_le_norm_f_norm_x) (simp!)+;
- finally;
- show "rabs (f x) <= function_norm E norm g * norm x"; .;
- qed;
-
- with f_norm f_cont; show "?R <= ?L";
- proof (rule fnorm_le_ub);
+ proof (rule norm_fx_le_norm_f_norm_x);
+ show "x:E"; ..;
+ qed;
+ finally; show "rabs (f x) <= function_norm E norm g * norm x"; .;
+ qed;
+ thus "?R <= ?L";
+ proof (rule fnorm_le_ub [OF f_norm f_cont]);
from g_cont; show "0r <= function_norm E norm g"; ..;
qed;
qed;