src/HOL/Hahn_Banach/Hahn_Banach.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
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modernized header uniformly as section;
     1 (*  Title:      HOL/Hahn_Banach/Hahn_Banach.thy
     2     Author:     Gertrud Bauer, TU Munich
     3 *)
     4 
     5 section \<open>The Hahn-Banach Theorem\<close>
     6 
     7 theory Hahn_Banach
     8 imports Hahn_Banach_Lemmas
     9 begin
    10 
    11 text \<open>
    12   We present the proof of two different versions of the Hahn-Banach
    13   Theorem, closely following @{cite \<open>\S36\<close> "Heuser:1986"}.
    14 \<close>
    15 
    16 subsection \<open>The Hahn-Banach Theorem for vector spaces\<close>
    17 
    18 text \<open>
    19   \textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real
    20   vector space @{text E}, let @{text p} be a semi-norm on @{text E},
    21   and @{text f} be a linear form defined on @{text F} such that @{text
    22   f} is bounded by @{text p}, i.e.  @{text "\<forall>x \<in> F. f x \<le> p x"}.  Then
    23   @{text f} can be extended to a linear form @{text h} on @{text E}
    24   such that @{text h} is norm-preserving, i.e. @{text h} is also
    25   bounded by @{text p}.
    26 
    27   \bigskip
    28   \textbf{Proof Sketch.}
    29   \begin{enumerate}
    30 
    31   \item Define @{text M} as the set of norm-preserving extensions of
    32   @{text f} to subspaces of @{text E}. The linear forms in @{text M}
    33   are ordered by domain extension.
    34 
    35   \item We show that every non-empty chain in @{text M} has an upper
    36   bound in @{text M}.
    37 
    38   \item With Zorn's Lemma we conclude that there is a maximal function
    39   @{text g} in @{text M}.
    40 
    41   \item The domain @{text H} of @{text g} is the whole space @{text
    42   E}, as shown by classical contradiction:
    43 
    44   \begin{itemize}
    45 
    46   \item Assuming @{text g} is not defined on whole @{text E}, it can
    47   still be extended in a norm-preserving way to a super-space @{text
    48   H'} of @{text H}.
    49 
    50   \item Thus @{text g} can not be maximal. Contradiction!
    51 
    52   \end{itemize}
    53   \end{enumerate}
    54 \<close>
    55 
    56 theorem Hahn_Banach:
    57   assumes E: "vectorspace E" and "subspace F E"
    58     and "seminorm E p" and "linearform F f"
    59   assumes fp: "\<forall>x \<in> F. f x \<le> p x"
    60   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)"
    61     -- \<open>Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E},\<close>
    62     -- \<open>and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p},\<close>
    63     -- \<open>then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \skp\<close>
    64 proof -
    65   interpret vectorspace E by fact
    66   interpret subspace F E by fact
    67   interpret seminorm E p by fact
    68   interpret linearform F f by fact
    69   def M \<equiv> "norm_pres_extensions E p F f"
    70   then have M: "M = \<dots>" by (simp only:)
    71   from E have F: "vectorspace F" ..
    72   note FE = \<open>F \<unlhd> E\<close>
    73   {
    74     fix c assume cM: "c \<in> chains M" and ex: "\<exists>x. x \<in> c"
    75     have "\<Union>c \<in> M"
    76       -- \<open>Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}:\<close>
    77       -- \<open>@{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}.\<close>
    78       unfolding M_def
    79     proof (rule norm_pres_extensionI)
    80       let ?H = "domain (\<Union>c)"
    81       let ?h = "funct (\<Union>c)"
    82 
    83       have a: "graph ?H ?h = \<Union>c"
    84       proof (rule graph_domain_funct)
    85         fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"
    86         with M_def cM show "z = y" by (rule sup_definite)
    87       qed
    88       moreover from M cM a have "linearform ?H ?h"
    89         by (rule sup_lf)
    90       moreover from a M cM ex FE E have "?H \<unlhd> E"
    91         by (rule sup_subE)
    92       moreover from a M cM ex FE have "F \<unlhd> ?H"
    93         by (rule sup_supF)
    94       moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
    95         by (rule sup_ext)
    96       moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"
    97         by (rule sup_norm_pres)
    98       ultimately show "\<exists>H h. \<Union>c = graph H h
    99           \<and> linearform H h
   100           \<and> H \<unlhd> E
   101           \<and> F \<unlhd> H
   102           \<and> graph F f \<subseteq> graph H h
   103           \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast
   104     qed
   105   }
   106   then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"
   107   -- \<open>With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp\<close>
   108   proof (rule Zorn's_Lemma)
   109       -- \<open>We show that @{text M} is non-empty:\<close>
   110     show "graph F f \<in> M"
   111       unfolding M_def
   112     proof (rule norm_pres_extensionI2)
   113       show "linearform F f" by fact
   114       show "F \<unlhd> E" by fact
   115       from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)
   116       show "graph F f \<subseteq> graph F f" ..
   117       show "\<forall>x\<in>F. f x \<le> p x" by fact
   118     qed
   119   qed
   120   then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
   121     by blast
   122   from gM obtain H h where
   123       g_rep: "g = graph H h"
   124     and linearform: "linearform H h"
   125     and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"
   126     and graphs: "graph F f \<subseteq> graph H h"
   127     and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..
   128       -- \<open>@{text g} is a norm-preserving extension of @{text f}, in other words:\<close>
   129       -- \<open>@{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E},\<close>
   130       -- \<open>and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp\<close>
   131   from HE E have H: "vectorspace H"
   132     by (rule subspace.vectorspace)
   133 
   134   have HE_eq: "H = E"
   135     -- \<open>We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp\<close>
   136   proof (rule classical)
   137     assume neq: "H \<noteq> E"
   138       -- \<open>Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended\<close>
   139       -- \<open>in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp\<close>
   140     have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
   141     proof -
   142       from HE have "H \<subseteq> E" ..
   143       with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast
   144       obtain x': "x' \<noteq> 0"
   145       proof
   146         show "x' \<noteq> 0"
   147         proof
   148           assume "x' = 0"
   149           with H have "x' \<in> H" by (simp only: vectorspace.zero)
   150           with \<open>x' \<notin> H\<close> show False by contradiction
   151         qed
   152       qed
   153 
   154       def H' \<equiv> "H + lin x'"
   155         -- \<open>Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp\<close>
   156       have HH': "H \<unlhd> H'"
   157       proof (unfold H'_def)
   158         from x'E have "vectorspace (lin x')" ..
   159         with H show "H \<unlhd> H + lin x'" ..
   160       qed
   161 
   162       obtain xi where
   163         xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
   164           \<and> xi \<le> p (y + x') - h y"
   165         -- \<open>Pick a real number @{text \<xi>} that fulfills certain inequations; this will\<close>
   166         -- \<open>be used to establish that @{text h'} is a norm-preserving extension of @{text h}.
   167            \label{ex-xi-use}\skp\<close>
   168       proof -
   169         from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
   170             \<and> xi \<le> p (y + x') - h y"
   171         proof (rule ex_xi)
   172           fix u v assume u: "u \<in> H" and v: "v \<in> H"
   173           with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto
   174           from H u v linearform have "h v - h u = h (v - u)"
   175             by (simp add: linearform.diff)
   176           also from hp and H u v have "\<dots> \<le> p (v - u)"
   177             by (simp only: vectorspace.diff_closed)
   178           also from x'E uE vE have "v - u = x' + - x' + v + - u"
   179             by (simp add: diff_eq1)
   180           also from x'E uE vE have "\<dots> = v + x' + - (u + x')"
   181             by (simp add: add_ac)
   182           also from x'E uE vE have "\<dots> = (v + x') - (u + x')"
   183             by (simp add: diff_eq1)
   184           also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"
   185             by (simp add: diff_subadditive)
   186           finally have "h v - h u \<le> p (v + x') + p (u + x')" .
   187           then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp
   188         qed
   189         then show thesis by (blast intro: that)
   190       qed
   191 
   192       def h' \<equiv> "\<lambda>x. let (y, a) =
   193           SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"
   194         -- \<open>Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp\<close>
   195 
   196       have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
   197         -- \<open>@{text h'} is an extension of @{text h} \dots \skp\<close>
   198       proof
   199         show "g \<subseteq> graph H' h'"
   200         proof -
   201           have  "graph H h \<subseteq> graph H' h'"
   202           proof (rule graph_extI)
   203             fix t assume t: "t \<in> H"
   204             from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
   205               using \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> by (rule decomp_H'_H)
   206             with h'_def show "h t = h' t" by (simp add: Let_def)
   207           next
   208             from HH' show "H \<subseteq> H'" ..
   209           qed
   210           with g_rep show ?thesis by (simp only:)
   211         qed
   212 
   213         show "g \<noteq> graph H' h'"
   214         proof -
   215           have "graph H h \<noteq> graph H' h'"
   216           proof
   217             assume eq: "graph H h = graph H' h'"
   218             have "x' \<in> H'"
   219               unfolding H'_def
   220             proof
   221               from H show "0 \<in> H" by (rule vectorspace.zero)
   222               from x'E show "x' \<in> lin x'" by (rule x_lin_x)
   223               from x'E show "x' = 0 + x'" by simp
   224             qed
   225             then have "(x', h' x') \<in> graph H' h'" ..
   226             with eq have "(x', h' x') \<in> graph H h" by (simp only:)
   227             then have "x' \<in> H" ..
   228             with \<open>x' \<notin> H\<close> show False by contradiction
   229           qed
   230           with g_rep show ?thesis by simp
   231         qed
   232       qed
   233       moreover have "graph H' h' \<in> M"
   234         -- \<open>and @{text h'} is norm-preserving. \skp\<close>
   235       proof (unfold M_def)
   236         show "graph H' h' \<in> norm_pres_extensions E p F f"
   237         proof (rule norm_pres_extensionI2)
   238           show "linearform H' h'"
   239             using h'_def H'_def HE linearform \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E
   240             by (rule h'_lf)
   241           show "H' \<unlhd> E"
   242           unfolding H'_def
   243           proof
   244             show "H \<unlhd> E" by fact
   245             show "vectorspace E" by fact
   246             from x'E show "lin x' \<unlhd> E" ..
   247           qed
   248           from H \<open>F \<unlhd> H\<close> HH' show FH': "F \<unlhd> H'"
   249             by (rule vectorspace.subspace_trans)
   250           show "graph F f \<subseteq> graph H' h'"
   251           proof (rule graph_extI)
   252             fix x assume x: "x \<in> F"
   253             with graphs have "f x = h x" ..
   254             also have "\<dots> = h x + 0 * xi" by simp
   255             also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"
   256               by (simp add: Let_def)
   257             also have "(x, 0) =
   258                 (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"
   259               using E HE
   260             proof (rule decomp_H'_H [symmetric])
   261               from FH x show "x \<in> H" ..
   262               from x' show "x' \<noteq> 0" .
   263               show "x' \<notin> H" by fact
   264               show "x' \<in> E" by fact
   265             qed
   266             also have
   267               "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)
   268               in h y + a * xi) = h' x" by (simp only: h'_def)
   269             finally show "f x = h' x" .
   270           next
   271             from FH' show "F \<subseteq> H'" ..
   272           qed
   273           show "\<forall>x \<in> H'. h' x \<le> p x"
   274             using h'_def H'_def \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E HE
   275               \<open>seminorm E p\<close> linearform and hp xi
   276             by (rule h'_norm_pres)
   277         qed
   278       qed
   279       ultimately show ?thesis ..
   280     qed
   281     then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
   282       -- \<open>So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp\<close>
   283     with gx show "H = E" by contradiction
   284   qed
   285 
   286   from HE_eq and linearform have "linearform E h"
   287     by (simp only:)
   288   moreover have "\<forall>x \<in> F. h x = f x"
   289   proof
   290     fix x assume "x \<in> F"
   291     with graphs have "f x = h x" ..
   292     then show "h x = f x" ..
   293   qed
   294   moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
   295     by (simp only:)
   296   ultimately show ?thesis by blast
   297 qed
   298 
   299 
   300 subsection  \<open>Alternative formulation\<close>
   301 
   302 text \<open>
   303   The following alternative formulation of the Hahn-Banach
   304   Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear
   305   form @{text f} and a seminorm @{text p} the following inequations
   306   are equivalent:\footnote{This was shown in lemma @{thm [source]
   307   abs_ineq_iff} (see page \pageref{abs-ineq-iff}).}
   308   \begin{center}
   309   \begin{tabular}{lll}
   310   @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
   311   @{text "\<forall>x \<in> H. h x \<le> p x"} \\
   312   \end{tabular}
   313   \end{center}
   314 \<close>
   315 
   316 theorem abs_Hahn_Banach:
   317   assumes E: "vectorspace E" and FE: "subspace F E"
   318     and lf: "linearform F f" and sn: "seminorm E p"
   319   assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
   320   shows "\<exists>g. linearform E g
   321     \<and> (\<forall>x \<in> F. g x = f x)
   322     \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
   323 proof -
   324   interpret vectorspace E by fact
   325   interpret subspace F E by fact
   326   interpret linearform F f by fact
   327   interpret seminorm E p by fact
   328   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)"
   329     using E FE sn lf
   330   proof (rule Hahn_Banach)
   331     show "\<forall>x \<in> F. f x \<le> p x"
   332       using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])
   333   qed
   334   then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"
   335       and **: "\<forall>x \<in> E. g x \<le> p x" by blast
   336   have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
   337     using _ E sn lg **
   338   proof (rule abs_ineq_iff [THEN iffD2])
   339     show "E \<unlhd> E" ..
   340   qed
   341   with lg * show ?thesis by blast
   342 qed
   343 
   344 
   345 subsection \<open>The Hahn-Banach Theorem for normed spaces\<close>
   346 
   347 text \<open>
   348   Every continuous linear form @{text f} on a subspace @{text F} of a
   349   norm space @{text E}, can be extended to a continuous linear form
   350   @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}.
   351 \<close>
   352 
   353 theorem norm_Hahn_Banach:
   354   fixes V and norm ("\<parallel>_\<parallel>")
   355   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}"
   356   fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
   357   defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
   358   assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"
   359     and linearform: "linearform F f" and "continuous F f norm"
   360   shows "\<exists>g. linearform E g
   361      \<and> continuous E g norm
   362      \<and> (\<forall>x \<in> F. g x = f x)
   363      \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
   364 proof -
   365   interpret normed_vectorspace E norm by fact
   366   interpret normed_vectorspace_with_fn_norm E norm B fn_norm
   367     by (auto simp: B_def fn_norm_def) intro_locales
   368   interpret subspace F E by fact
   369   interpret linearform F f by fact
   370   interpret continuous F f norm by fact
   371   have E: "vectorspace E" by intro_locales
   372   have F: "vectorspace F" by rule intro_locales
   373   have F_norm: "normed_vectorspace F norm"
   374     using FE E_norm by (rule subspace_normed_vs)
   375   have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
   376     by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero
   377       [OF normed_vectorspace_with_fn_norm.intro,
   378        OF F_norm \<open>continuous F f norm\<close> , folded B_def fn_norm_def])
   379   txt \<open>We define a function @{text p} on @{text E} as follows:
   380     @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}\<close>
   381   def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
   382 
   383   txt \<open>@{text p} is a seminorm on @{text E}:\<close>
   384   have q: "seminorm E p"
   385   proof
   386     fix x y a assume x: "x \<in> E" and y: "y \<in> E"
   387     
   388     txt \<open>@{text p} is positive definite:\<close>
   389     have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
   390     moreover from x have "0 \<le> \<parallel>x\<parallel>" ..
   391     ultimately show "0 \<le> p x"  
   392       by (simp add: p_def zero_le_mult_iff)
   393 
   394     txt \<open>@{text p} is absolutely homogeneous:\<close>
   395 
   396     show "p (a \<cdot> x) = \<bar>a\<bar> * p x"
   397     proof -
   398       have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)
   399       also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)
   400       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
   401       also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)
   402       finally show ?thesis .
   403     qed
   404 
   405     txt \<open>Furthermore, @{text p} is subadditive:\<close>
   406 
   407     show "p (x + y) \<le> p x + p y"
   408     proof -
   409       have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)
   410       also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
   411       from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
   412       with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
   413         by (simp add: mult_left_mono)
   414       also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left)
   415       also have "\<dots> = p x + p y" by (simp only: p_def)
   416       finally show ?thesis .
   417     qed
   418   qed
   419 
   420   txt \<open>@{text f} is bounded by @{text p}.\<close>
   421 
   422   have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
   423   proof
   424     fix x assume "x \<in> F"
   425     with \<open>continuous F f norm\<close> and linearform
   426     show "\<bar>f x\<bar> \<le> p x"
   427       unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong
   428         [OF normed_vectorspace_with_fn_norm.intro,
   429          OF F_norm, folded B_def fn_norm_def])
   430   qed
   431 
   432   txt \<open>Using the fact that @{text p} is a seminorm and @{text f} is bounded
   433     by @{text p} we can apply the Hahn-Banach Theorem for real vector
   434     spaces. So @{text f} can be extended in a norm-preserving way to
   435     some function @{text g} on the whole vector space @{text E}.\<close>
   436 
   437   with E FE linearform q obtain g where
   438       linearformE: "linearform E g"
   439     and a: "\<forall>x \<in> F. g x = f x"
   440     and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
   441     by (rule abs_Hahn_Banach [elim_format]) iprover
   442 
   443   txt \<open>We furthermore have to show that @{text g} is also continuous:\<close>
   444 
   445   have g_cont: "continuous E g norm" using linearformE
   446   proof
   447     fix x assume "x \<in> E"
   448     with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
   449       by (simp only: p_def)
   450   qed
   451 
   452   txt \<open>To complete the proof, we show that @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}.\<close>
   453 
   454   have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
   455   proof (rule order_antisym)
   456     txt \<open>
   457       First we show @{text "\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>"}.  The function norm @{text
   458       "\<parallel>g\<parallel>"} is defined as the smallest @{text "c \<in> \<real>"} such that
   459       \begin{center}
   460       \begin{tabular}{l}
   461       @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
   462       \end{tabular}
   463       \end{center}
   464       \noindent Furthermore holds
   465       \begin{center}
   466       \begin{tabular}{l}
   467       @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
   468       \end{tabular}
   469       \end{center}
   470 \<close>
   471 
   472     from g_cont _ ge_zero
   473     show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
   474     proof
   475       fix x assume "x \<in> E"
   476       with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
   477         by (simp only: p_def)
   478     qed
   479 
   480     txt \<open>The other direction is achieved by a similar argument.\<close>
   481 
   482     show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
   483     proof (rule normed_vectorspace_with_fn_norm.fn_norm_least
   484         [OF normed_vectorspace_with_fn_norm.intro,
   485          OF F_norm, folded B_def fn_norm_def])
   486       fix x assume x: "x \<in> F"
   487       show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
   488       proof -
   489         from a x have "g x = f x" ..
   490         then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
   491         also from g_cont
   492         have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
   493         proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])
   494           from FE x show "x \<in> E" ..
   495         qed
   496         finally show ?thesis .
   497       qed
   498     next
   499       show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
   500         using g_cont
   501         by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
   502       show "continuous F f norm" by fact
   503     qed
   504   qed
   505   with linearformE a g_cont show ?thesis by blast
   506 qed
   507 
   508 end