--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Wed Aug 02 19:40:14 2000 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Thu Aug 03 00:34:22 2000 +0200
@@ -64,37 +64,37 @@
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)"
+ 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"
+ 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"
+ 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)"
+ 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)"
+ let ?H = "domain (\<Union>c)"
+ let ?h = "funct (\<Union>c)"
- show a: "graph ?H ?h = \\<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"
+ 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"
@@ -103,40 +103,40 @@
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"
+ show "graph F f \<subseteq> graph ?H ?h"
by (rule sup_ext, rule a) (simp!)+
- show "\\<forall>x \\<in> ?H. ?h x <= p x"
+ 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"
+ 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"
+ 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!)
+ 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"
+ 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 *}
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"
+ "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)"
+ 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
@@ -144,39 +144,39 @@
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 \<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'"
+ have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
proof -
- obtain x' where "x' \\<in> E" "x' \\<notin> H"
+ 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"
+ have "\<exists>x' \<in> E. x' \<notin> H"
proof (rule set_less_imp_diff_not_empty)
- have "H \\<subseteq> E" ..
- thus "H \\<subset> E" ..
+ have "H \<subseteq> E" ..
+ thus "H \<subset> E" ..
qed
thus ?thesis by blast
qed
- have x': "x' \\<noteq> 0"
+ have x': "x' \<noteq> 0"
proof (rule classical)
presume "x' = 0"
- with h have "x' \\<in> H" by simp
+ 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 *}
- obtain xi where "\\<forall>y \\<in> H. - p (y + x') - h y <= xi
- \\<and> xi <= p (y + x') - h y"
+ 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"
+ 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"
+ 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)"
@@ -197,25 +197,25 @@
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
+ 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 "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'"
+ show "g \<subseteq> graph H' h'"
proof -
- have "graph H h \\<subseteq> graph H' h'"
+ 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)
+ 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'"
+ show "H \<subseteq> H'"
proof (rule subspace_subset)
show "is_subspace H H'"
proof (unfold H'_def, rule subspace_vs_sum1)
@@ -226,29 +226,29 @@
qed
thus ?thesis by (simp!)
qed
- show "g \\<noteq> graph H' h'"
+ show "g \<noteq> graph H' h'"
proof -
- have "graph H h \\<noteq> graph H' h'"
+ have "graph H h \<noteq> graph H' h'"
proof
assume e: "graph H h = graph H' h'"
- have "x' \\<in> 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)
+ 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" ..
+ 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"
+ 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"
+ 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')+
@@ -261,47 +261,47 @@
finally (subspace_trans [OF _ h])
show f_h': "is_subspace F H'" .
- show "graph F f \\<subseteq> graph H' h'"
+ show "graph F f \<subseteq> graph H' h'"
proof (rule graph_extI)
- fix x assume "x \\<in> F"
+ 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)"
+ "(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)
+ "(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'" ..
+ from f_h' show "F \<subseteq> H'" ..
qed
- show "\\<forall>x \\<in> H'. h' x <= p x"
+ 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!)
+ thus "graph H' h' \<in> M" by (simp!)
qed
qed
qed
- hence "\\<not> (\\<forall>x \\<in> M. g \\<subseteq> x --> g = x)" by simp
+ 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)"
+ 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"
+ show "\<forall>x \<in> F. h x = f x"
proof
- fix x assume "x \\<in> F" have "f x = h x " ..
+ fix x assume "x \<in> F" have "f x = h x " ..
thus "h x = f x" ..
qed
- from eq show "\\<forall>x \\<in> E. h x <= p x" by (force!)
+ from eq show "\<forall>x \<in> E. h x <= p x" by (force!)
qed
qed
qed
@@ -322,21 +322,21 @@
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)"
+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)"
+"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"
+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
@@ -351,10 +351,10 @@
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"
+==> \<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"
@@ -368,32 +368,32 @@
\end{matharray}
*}
-def p == "\\<lambda>x. \\<parallel>f\\<parallel>F,norm * norm x"
+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"
+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" ..
+ 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"
+show "p (a \<cdot> x) = |a| * p x"
proof -
- have "p (a \\<cdot> x) = \\<parallel>f\\<parallel>F,norm * norm (a \\<cdot> x)"
+ 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"
+ 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)"
+ 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 .
@@ -403,16 +403,16 @@
show "p (x + y) <= p x + p y"
proof -
- have "p (x + y) = \\<parallel>f\\<parallel>F,norm * norm (x + y)"
+ 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)"
+ 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" ..
+ 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"
+ 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
@@ -420,9 +420,9 @@
txt{* $f$ is bounded by $p$. *}
-have "\\<forall>x \\<in> F. |f x| <= p x"
+have "\<forall>x \<in> F. |f x| <= p x"
proof
-fix x assume "x \\<in> F"
+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
@@ -434,20 +434,20 @@
$g$ 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)"
+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"
+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"
+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
@@ -455,15 +455,15 @@
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"
+ 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"
+ show "\<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
(is "?L = ?R")
proof (rule order_antisym)
@@ -478,36 +478,36 @@
\end{matharray}
*}
- have "\\<forall>x \\<in> E. |g x| <= \\<parallel>f\\<parallel>F,norm * norm x"
+ 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"
+ 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" ..
+ 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"
+ have "\<forall>x \<in> F. |f x| <= \<parallel>g\<parallel>E,norm * norm x"
proof
- fix x assume "x \\<in> F"
+ 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"
+ have "... <= \<parallel>g\<parallel>E,norm * norm x"
proof (rule norm_fx_le_norm_f_norm_x)
- show "x \\<in> E" ..
+ show "x \<in> E" ..
qed
- finally show "|f x| <= \\<parallel>g\\<parallel>E,norm * norm x" .
+ 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" ..
+ from g_cont show "#0 <= \<parallel>g\<parallel>E,norm" ..
qed
qed
qed