src/HOL/Hahn_Banach/Hahn_Banach.thy
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     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 Theorem,

    13   closely following @{cite \<open>\S36\<close> "Heuser:1986"}.

    14 \<close>

    15

    16

    17 subsection \<open>The Hahn-Banach Theorem for vector spaces\<close>

    18

    19 paragraph \<open>Hahn-Banach Theorem.\<close>

    20 text \<open>

    21   Let \<open>F\<close> be a subspace of a real vector space \<open>E\<close>, let \<open>p\<close> be a semi-norm on

    22   \<open>E\<close>, and \<open>f\<close> be a linear form defined on \<open>F\<close> such that \<open>f\<close> is bounded by

    23   \<open>p\<close>, i.e. \<open>\<forall>x \<in> F. f x \<le> p x\<close>. Then \<open>f\<close> can be extended to a linear form \<open>h\<close>

    24   on \<open>E\<close> such that \<open>h\<close> is norm-preserving, i.e. \<open>h\<close> is also bounded by \<open>p\<close>.

    25 \<close>

    26

    27 paragraph \<open>Proof Sketch.\<close>

    28 text \<open>

    29   \<^enum> Define \<open>M\<close> as the set of norm-preserving extensions of \<open>f\<close> to subspaces of

    30   \<open>E\<close>. The linear forms in \<open>M\<close> are ordered by domain extension.

    31

    32   \<^enum> We show that every non-empty chain in \<open>M\<close> has an upper bound in \<open>M\<close>.

    33

    34   \<^enum> With Zorn's Lemma we conclude that there is a maximal function \<open>g\<close> in \<open>M\<close>.

    35

    36   \<^enum> The domain \<open>H\<close> of \<open>g\<close> is the whole space \<open>E\<close>, as shown by classical

    37   contradiction:

    38

    39     \<^item> Assuming \<open>g\<close> is not defined on whole \<open>E\<close>, it can still be extended in a

    40     norm-preserving way to a super-space \<open>H'\<close> of \<open>H\<close>.

    41

    42     \<^item> Thus \<open>g\<close> can not be maximal. Contradiction!

    43 \<close>

    44

    45 theorem Hahn_Banach:

    46   assumes E: "vectorspace E" and "subspace F E"

    47     and "seminorm E p" and "linearform F f"

    48   assumes fp: "\<forall>x \<in> F. f x \<le> p x"

    49   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)"

    50     \<comment> \<open>Let \<open>E\<close> be a vector space, \<open>F\<close> a subspace of \<open>E\<close>, \<open>p\<close> a seminorm on \<open>E\<close>,\<close>

    51     \<comment> \<open>and \<open>f\<close> a linear form on \<open>F\<close> such that \<open>f\<close> is bounded by \<open>p\<close>,\<close>

    52     \<comment> \<open>then \<open>f\<close> can be extended to a linear form \<open>h\<close> on \<open>E\<close> in a norm-preserving way. \<^smallskip>\<close>

    53 proof -

    54   interpret vectorspace E by fact

    55   interpret subspace F E by fact

    56   interpret seminorm E p by fact

    57   interpret linearform F f by fact

    58   define M where "M = norm_pres_extensions E p F f"

    59   then have M: "M = \<dots>" by (simp only:)

    60   from E have F: "vectorspace F" ..

    61   note FE = \<open>F \<unlhd> E\<close>

    62   {

    63     fix c assume cM: "c \<in> chains M" and ex: "\<exists>x. x \<in> c"

    64     have "\<Union>c \<in> M"

    65       \<comment> \<open>Show that every non-empty chain \<open>c\<close> of \<open>M\<close> has an upper bound in \<open>M\<close>:\<close>

    66       \<comment> \<open>\<open>\<Union>c\<close> is greater than any element of the chain \<open>c\<close>, so it suffices to show \<open>\<Union>c \<in> M\<close>.\<close>

    67       unfolding M_def

    68     proof (rule norm_pres_extensionI)

    69       let ?H = "domain (\<Union>c)"

    70       let ?h = "funct (\<Union>c)"

    71

    72       have a: "graph ?H ?h = \<Union>c"

    73       proof (rule graph_domain_funct)

    74         fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"

    75         with M_def cM show "z = y" by (rule sup_definite)

    76       qed

    77       moreover from M cM a have "linearform ?H ?h"

    78         by (rule sup_lf)

    79       moreover from a M cM ex FE E have "?H \<unlhd> E"

    80         by (rule sup_subE)

    81       moreover from a M cM ex FE have "F \<unlhd> ?H"

    82         by (rule sup_supF)

    83       moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"

    84         by (rule sup_ext)

    85       moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"

    86         by (rule sup_norm_pres)

    87       ultimately show "\<exists>H h. \<Union>c = graph H h

    88           \<and> linearform H h

    89           \<and> H \<unlhd> E

    90           \<and> F \<unlhd> H

    91           \<and> graph F f \<subseteq> graph H h

    92           \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast

    93     qed

    94   }

    95   then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"

    96   \<comment> \<open>With Zorn's Lemma we can conclude that there is a maximal element in \<open>M\<close>. \<^smallskip>\<close>

    97   proof (rule Zorn's_Lemma)

    98       \<comment> \<open>We show that \<open>M\<close> is non-empty:\<close>

    99     show "graph F f \<in> M"

   100       unfolding M_def

   101     proof (rule norm_pres_extensionI2)

   102       show "linearform F f" by fact

   103       show "F \<unlhd> E" by fact

   104       from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)

   105       show "graph F f \<subseteq> graph F f" ..

   106       show "\<forall>x\<in>F. f x \<le> p x" by fact

   107     qed

   108   qed

   109   then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"

   110     by blast

   111   from gM obtain H h where

   112       g_rep: "g = graph H h"

   113     and linearform: "linearform H h"

   114     and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"

   115     and graphs: "graph F f \<subseteq> graph H h"

   116     and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..

   117       \<comment> \<open>\<open>g\<close> is a norm-preserving extension of \<open>f\<close>, in other words:\<close>

   118       \<comment> \<open>\<open>g\<close> is the graph of some linear form \<open>h\<close> defined on a subspace \<open>H\<close> of \<open>E\<close>,\<close>

   119       \<comment> \<open>and \<open>h\<close> is an extension of \<open>f\<close> that is again bounded by \<open>p\<close>. \<^smallskip>\<close>

   120   from HE E have H: "vectorspace H"

   121     by (rule subspace.vectorspace)

   122

   123   have HE_eq: "H = E"

   124     \<comment> \<open>We show that \<open>h\<close> is defined on whole \<open>E\<close> by classical contradiction. \<^smallskip>\<close>

   125   proof (rule classical)

   126     assume neq: "H \<noteq> E"

   127       \<comment> \<open>Assume \<open>h\<close> is not defined on whole \<open>E\<close>. Then show that \<open>h\<close> can be extended\<close>

   128       \<comment> \<open>in a norm-preserving way to a function \<open>h'\<close> with the graph \<open>g'\<close>. \<^smallskip>\<close>

   129     have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"

   130     proof -

   131       from HE have "H \<subseteq> E" ..

   132       with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast

   133       obtain x': "x' \<noteq> 0"

   134       proof

   135         show "x' \<noteq> 0"

   136         proof

   137           assume "x' = 0"

   138           with H have "x' \<in> H" by (simp only: vectorspace.zero)

   139           with \<open>x' \<notin> H\<close> show False by contradiction

   140         qed

   141       qed

   142

   143       define H' where "H' = H + lin x'"

   144         \<comment> \<open>Define \<open>H'\<close> as the direct sum of \<open>H\<close> and the linear closure of \<open>x'\<close>. \<^smallskip>\<close>

   145       have HH': "H \<unlhd> H'"

   146       proof (unfold H'_def)

   147         from x'E have "vectorspace (lin x')" ..

   148         with H show "H \<unlhd> H + lin x'" ..

   149       qed

   150

   151       obtain xi where

   152         xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi

   153           \<and> xi \<le> p (y + x') - h y"

   154         \<comment> \<open>Pick a real number \<open>\<xi>\<close> that fulfills certain inequality; this will\<close>

   155         \<comment> \<open>be used to establish that \<open>h'\<close> is a norm-preserving extension of \<open>h\<close>.

   156            \label{ex-xi-use}\<^smallskip>\<close>

   157       proof -

   158         from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi

   159             \<and> xi \<le> p (y + x') - h y"

   160         proof (rule ex_xi)

   161           fix u v assume u: "u \<in> H" and v: "v \<in> H"

   162           with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto

   163           from H u v linearform have "h v - h u = h (v - u)"

   164             by (simp add: linearform.diff)

   165           also from hp and H u v have "\<dots> \<le> p (v - u)"

   166             by (simp only: vectorspace.diff_closed)

   167           also from x'E uE vE have "v - u = x' + - x' + v + - u"

   168             by (simp add: diff_eq1)

   169           also from x'E uE vE have "\<dots> = v + x' + - (u + x')"

   170             by (simp add: add_ac)

   171           also from x'E uE vE have "\<dots> = (v + x') - (u + x')"

   172             by (simp add: diff_eq1)

   173           also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"

   174             by (simp add: diff_subadditive)

   175           finally have "h v - h u \<le> p (v + x') + p (u + x')" .

   176           then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp

   177         qed

   178         then show thesis by (blast intro: that)

   179       qed

   180

   181       define h' where "h' x = (let (y, a) =

   182           SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi)" for x

   183         \<comment> \<open>Define the extension \<open>h'\<close> of \<open>h\<close> to \<open>H'\<close> using \<open>\<xi>\<close>. \<^smallskip>\<close>

   184

   185       have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"

   186         \<comment> \<open>\<open>h'\<close> is an extension of \<open>h\<close> \dots \<^smallskip>\<close>

   187       proof

   188         show "g \<subseteq> graph H' h'"

   189         proof -

   190           have "graph H h \<subseteq> graph H' h'"

   191           proof (rule graph_extI)

   192             fix t assume t: "t \<in> H"

   193             from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"

   194               using \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> by (rule decomp_H'_H)

   195             with h'_def show "h t = h' t" by (simp add: Let_def)

   196           next

   197             from HH' show "H \<subseteq> H'" ..

   198           qed

   199           with g_rep show ?thesis by (simp only:)

   200         qed

   201

   202         show "g \<noteq> graph H' h'"

   203         proof -

   204           have "graph H h \<noteq> graph H' h'"

   205           proof

   206             assume eq: "graph H h = graph H' h'"

   207             have "x' \<in> H'"

   208               unfolding H'_def

   209             proof

   210               from H show "0 \<in> H" by (rule vectorspace.zero)

   211               from x'E show "x' \<in> lin x'" by (rule x_lin_x)

   212               from x'E show "x' = 0 + x'" by simp

   213             qed

   214             then have "(x', h' x') \<in> graph H' h'" ..

   215             with eq have "(x', h' x') \<in> graph H h" by (simp only:)

   216             then have "x' \<in> H" ..

   217             with \<open>x' \<notin> H\<close> show False by contradiction

   218           qed

   219           with g_rep show ?thesis by simp

   220         qed

   221       qed

   222       moreover have "graph H' h' \<in> M"

   223         \<comment> \<open>and \<open>h'\<close> is norm-preserving. \<^smallskip>\<close>

   224       proof (unfold M_def)

   225         show "graph H' h' \<in> norm_pres_extensions E p F f"

   226         proof (rule norm_pres_extensionI2)

   227           show "linearform H' h'"

   228             using h'_def H'_def HE linearform \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E

   229             by (rule h'_lf)

   230           show "H' \<unlhd> E"

   231           unfolding H'_def

   232           proof

   233             show "H \<unlhd> E" by fact

   234             show "vectorspace E" by fact

   235             from x'E show "lin x' \<unlhd> E" ..

   236           qed

   237           from H \<open>F \<unlhd> H\<close> HH' show FH': "F \<unlhd> H'"

   238             by (rule vectorspace.subspace_trans)

   239           show "graph F f \<subseteq> graph H' h'"

   240           proof (rule graph_extI)

   241             fix x assume x: "x \<in> F"

   242             with graphs have "f x = h x" ..

   243             also have "\<dots> = h x + 0 * xi" by simp

   244             also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"

   245               by (simp add: Let_def)

   246             also have "(x, 0) =

   247                 (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"

   248               using E HE

   249             proof (rule decomp_H'_H [symmetric])

   250               from FH x show "x \<in> H" ..

   251               from x' show "x' \<noteq> 0" .

   252               show "x' \<notin> H" by fact

   253               show "x' \<in> E" by fact

   254             qed

   255             also have

   256               "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)

   257               in h y + a * xi) = h' x" by (simp only: h'_def)

   258             finally show "f x = h' x" .

   259           next

   260             from FH' show "F \<subseteq> H'" ..

   261           qed

   262           show "\<forall>x \<in> H'. h' x \<le> p x"

   263             using h'_def H'_def \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E HE

   264               \<open>seminorm E p\<close> linearform and hp xi

   265             by (rule h'_norm_pres)

   266         qed

   267       qed

   268       ultimately show ?thesis ..

   269     qed

   270     then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp

   271       \<comment> \<open>So the graph \<open>g\<close> of \<open>h\<close> cannot be maximal. Contradiction! \<^smallskip>\<close>

   272     with gx show "H = E" by contradiction

   273   qed

   274

   275   from HE_eq and linearform have "linearform E h"

   276     by (simp only:)

   277   moreover have "\<forall>x \<in> F. h x = f x"

   278   proof

   279     fix x assume "x \<in> F"

   280     with graphs have "f x = h x" ..

   281     then show "h x = f x" ..

   282   qed

   283   moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"

   284     by (simp only:)

   285   ultimately show ?thesis by blast

   286 qed

   287

   288

   289 subsection \<open>Alternative formulation\<close>

   290

   291 text \<open>

   292   The following alternative formulation of the Hahn-Banach

   293   Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear form \<open>f\<close>

   294   and a seminorm \<open>p\<close> the following inequality are equivalent:\footnote{This

   295   was shown in lemma @{thm [source] abs_ineq_iff} (see page

   296   \pageref{abs-ineq-iff}).}

   297   \begin{center}

   298   \begin{tabular}{lll}

   299   \<open>\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x\<close> & and & \<open>\<forall>x \<in> H. h x \<le> p x\<close> \\

   300   \end{tabular}

   301   \end{center}

   302 \<close>

   303

   304 theorem abs_Hahn_Banach:

   305   assumes E: "vectorspace E" and FE: "subspace F E"

   306     and lf: "linearform F f" and sn: "seminorm E p"

   307   assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

   308   shows "\<exists>g. linearform E g

   309     \<and> (\<forall>x \<in> F. g x = f x)

   310     \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"

   311 proof -

   312   interpret vectorspace E by fact

   313   interpret subspace F E by fact

   314   interpret linearform F f by fact

   315   interpret seminorm E p by fact

   316   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)"

   317     using E FE sn lf

   318   proof (rule Hahn_Banach)

   319     show "\<forall>x \<in> F. f x \<le> p x"

   320       using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])

   321   qed

   322   then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"

   323       and **: "\<forall>x \<in> E. g x \<le> p x" by blast

   324   have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

   325     using _ E sn lg **

   326   proof (rule abs_ineq_iff [THEN iffD2])

   327     show "E \<unlhd> E" ..

   328   qed

   329   with lg * show ?thesis by blast

   330 qed

   331

   332

   333 subsection \<open>The Hahn-Banach Theorem for normed spaces\<close>

   334

   335 text \<open>

   336   Every continuous linear form \<open>f\<close> on a subspace \<open>F\<close> of a norm space \<open>E\<close>, can

   337   be extended to a continuous linear form \<open>g\<close> on \<open>E\<close> such that \<open>\<parallel>f\<parallel> = \<parallel>g\<parallel>\<close>.

   338 \<close>

   339

   340 theorem norm_Hahn_Banach:

   341   fixes V and norm ("\<parallel>_\<parallel>")

   342   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}"

   343   fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)

   344   defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"

   345   assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"

   346     and linearform: "linearform F f" and "continuous F f norm"

   347   shows "\<exists>g. linearform E g

   348      \<and> continuous E g norm

   349      \<and> (\<forall>x \<in> F. g x = f x)

   350      \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

   351 proof -

   352   interpret normed_vectorspace E norm by fact

   353   interpret normed_vectorspace_with_fn_norm E norm B fn_norm

   354     by (auto simp: B_def fn_norm_def) intro_locales

   355   interpret subspace F E by fact

   356   interpret linearform F f by fact

   357   interpret continuous F f norm by fact

   358   have E: "vectorspace E" by intro_locales

   359   have F: "vectorspace F" by rule intro_locales

   360   have F_norm: "normed_vectorspace F norm"

   361     using FE E_norm by (rule subspace_normed_vs)

   362   have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"

   363     by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero

   364       [OF normed_vectorspace_with_fn_norm.intro,

   365        OF F_norm \<open>continuous F f norm\<close> , folded B_def fn_norm_def])

   366   txt \<open>We define a function \<open>p\<close> on \<open>E\<close> as follows:

   367     \<open>p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>\<close>

   368   define p where "p x = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>" for x

   369

   370   txt \<open>\<open>p\<close> is a seminorm on \<open>E\<close>:\<close>

   371   have q: "seminorm E p"

   372   proof

   373     fix x y a assume x: "x \<in> E" and y: "y \<in> E"

   374

   375     txt \<open>\<open>p\<close> is positive definite:\<close>

   376     have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

   377     moreover from x have "0 \<le> \<parallel>x\<parallel>" ..

   378     ultimately show "0 \<le> p x"

   379       by (simp add: p_def zero_le_mult_iff)

   380

   381     txt \<open>\<open>p\<close> is absolutely homogeneous:\<close>

   382

   383     show "p (a \<cdot> x) = \<bar>a\<bar> * p x"

   384     proof -

   385       have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)

   386       also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)

   387       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

   388       also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)

   389       finally show ?thesis .

   390     qed

   391

   392     txt \<open>Furthermore, \<open>p\<close> is subadditive:\<close>

   393

   394     show "p (x + y) \<le> p x + p y"

   395     proof -

   396       have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)

   397       also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

   398       from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..

   399       with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"

   400         by (simp add: mult_left_mono)

   401       also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left)

   402       also have "\<dots> = p x + p y" by (simp only: p_def)

   403       finally show ?thesis .

   404     qed

   405   qed

   406

   407   txt \<open>\<open>f\<close> is bounded by \<open>p\<close>.\<close>

   408

   409   have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

   410   proof

   411     fix x assume "x \<in> F"

   412     with \<open>continuous F f norm\<close> and linearform

   413     show "\<bar>f x\<bar> \<le> p x"

   414       unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong

   415         [OF normed_vectorspace_with_fn_norm.intro,

   416          OF F_norm, folded B_def fn_norm_def])

   417   qed

   418

   419   txt \<open>Using the fact that \<open>p\<close> is a seminorm and \<open>f\<close> is bounded by \<open>p\<close> we can

   420     apply the Hahn-Banach Theorem for real vector spaces. So \<open>f\<close> can be

   421     extended in a norm-preserving way to some function \<open>g\<close> on the whole vector

   422     space \<open>E\<close>.\<close>

   423

   424   with E FE linearform q obtain g where

   425       linearformE: "linearform E g"

   426     and a: "\<forall>x \<in> F. g x = f x"

   427     and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

   428     by (rule abs_Hahn_Banach [elim_format]) iprover

   429

   430   txt \<open>We furthermore have to show that \<open>g\<close> is also continuous:\<close>

   431

   432   have g_cont: "continuous E g norm" using linearformE

   433   proof

   434     fix x assume "x \<in> E"

   435     with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

   436       by (simp only: p_def)

   437   qed

   438

   439   txt \<open>To complete the proof, we show that \<open>\<parallel>g\<parallel> = \<parallel>f\<parallel>\<close>.\<close>

   440

   441   have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

   442   proof (rule order_antisym)

   443     txt \<open>

   444       First we show \<open>\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>\<close>. The function norm \<open>\<parallel>g\<parallel>\<close> is defined as the

   445       smallest \<open>c \<in> \<real>\<close> such that

   446       \begin{center}

   447       \begin{tabular}{l}

   448       \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>\<close>

   449       \end{tabular}

   450       \end{center}

   451       \<^noindent> Furthermore holds

   452       \begin{center}

   453       \begin{tabular}{l}

   454       \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>

   455       \end{tabular}

   456       \end{center}

   457     \<close>

   458

   459     from g_cont _ ge_zero

   460     show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"

   461     proof

   462       fix x assume "x \<in> E"

   463       with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

   464         by (simp only: p_def)

   465     qed

   466

   467     txt \<open>The other direction is achieved by a similar argument.\<close>

   468

   469     show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"

   470     proof (rule normed_vectorspace_with_fn_norm.fn_norm_least

   471         [OF normed_vectorspace_with_fn_norm.intro,

   472          OF F_norm, folded B_def fn_norm_def])

   473       fix x assume x: "x \<in> F"

   474       show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

   475       proof -

   476         from a x have "g x = f x" ..

   477         then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)

   478         also from g_cont have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

   479         proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])

   480           from FE x show "x \<in> E" ..

   481         qed

   482         finally show ?thesis .

   483       qed

   484     next

   485       show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"

   486         using g_cont by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])

   487       show "continuous F f norm" by fact

   488     qed

   489   qed

   490   with linearformE a g_cont show ?thesis by blast

   491 qed

   492

   493 end