--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Tue Dec 30 08:18:54 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,509 +0,0 @@
-(* Title: HOL/Real/HahnBanach/HahnBanach.thy
- Author: Gertrud Bauer, TU Munich
-*)
-
-header {* The Hahn-Banach Theorem *}
-
-theory HahnBanach
-imports HahnBanachLemmas
-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 HahnBanach:
- 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> chain 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> g = x"
- -- {* 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-HahnBanach} 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_HahnBanach:
- 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 HahnBanach)
- 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_HahnBanach:
- 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 norm f"
- shows "\<exists>g. linearform E g
- \<and> continuous E norm g
- \<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 norm f 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 norm f` , 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 homogenous: *}
-
- 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: right_distrib)
- 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 norm f` 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_HahnBanach [elim_format]) iprover
-
- txt {* We furthermore have to show that @{text g} is also continuous: *}
-
- have g_cont: "continuous E norm g" 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}
- *}
-
- have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
- 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
- from g_cont this ge_zero
- show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
- by (rule fn_norm_least [of g, folded B_def fn_norm_def])
-
- 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])
- show "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
- proof
- fix x assume x: "x \<in> F"
- 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 g, folded B_def fn_norm_def])
- from FE x show "x \<in> E" ..
- qed
- finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
- qed
- 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 norm f" by fact
- qed
- qed
- with linearformE a g_cont show ?thesis by blast
-qed
-
-end