src/HOL/Hahn_Banach/Hahn_Banach.thy
author wenzelm
Mon, 02 Nov 2015 16:02:32 +0100
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child 61583 c2b7033fa0ba
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(*  Title:      HOL/Hahn_Banach/Hahn_Banach.thy
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    Author:     Gertrud Bauer, TU Munich
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*)
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section \<open>The Hahn-Banach Theorem\<close>
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theory Hahn_Banach
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imports Hahn_Banach_Lemmas
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begin
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text \<open>
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  We present the proof of two different versions of the Hahn-Banach Theorem,
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  closely following @{cite \<open>\S36\<close> "Heuser:1986"}.
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\<close>
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subsection \<open>The Hahn-Banach Theorem for vector spaces\<close>
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paragraph \<open>Hahn-Banach Theorem.\<close>
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text \<open>
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  Let \<open>F\<close> be a subspace of a real vector space \<open>E\<close>, let \<open>p\<close> be a semi-norm
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  on \<open>E\<close>, and \<open>f\<close> be a linear form defined on \<open>F\<close> such that \<open>f\<close> is bounded
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  by \<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
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  form \<open>h\<close> on \<open>E\<close> such that \<open>h\<close> is norm-preserving, i.e. \<open>h\<close> is also bounded
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  by \<open>p\<close>.
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\<close>
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paragraph \<open>Proof Sketch.\<close>
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text \<open>
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  \<^enum> Define \<open>M\<close> as the set of norm-preserving extensions of \<open>f\<close> to subspaces
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  of \<open>E\<close>. The linear forms in \<open>M\<close> are ordered by domain extension.
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  \<^enum> We show that every non-empty chain in \<open>M\<close> has an upper bound in \<open>M\<close>.
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  \<^enum> With Zorn's Lemma we conclude that there is a maximal function \<open>g\<close> in
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  \<open>M\<close>.
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  \<^enum> The domain \<open>H\<close> of \<open>g\<close> is the whole space \<open>E\<close>, as shown by classical
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  contradiction:
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    \<^item> Assuming \<open>g\<close> is not defined on whole \<open>E\<close>, it can still be extended in
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    a norm-preserving way to a super-space \<open>H'\<close> of \<open>H\<close>.
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    \<^item> Thus \<open>g\<close> can not be maximal. Contradiction!
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\<close>
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theorem Hahn_Banach:
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  assumes E: "vectorspace E" and "subspace F E"
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    and "seminorm E p" and "linearform F f"
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  assumes fp: "\<forall>x \<in> F. f x \<le> p x"
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  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)"
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    -- \<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>
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    -- \<open>and \<open>f\<close> a linear form on \<open>F\<close> such that \<open>f\<close> is bounded by \<open>p\<close>,\<close>
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    -- \<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>
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proof -
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  interpret vectorspace E by fact
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  interpret subspace F E by fact
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  interpret seminorm E p by fact
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  interpret linearform F f by fact
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  def M \<equiv> "norm_pres_extensions E p F f"
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  then have M: "M = \<dots>" by (simp only:)
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  from E have F: "vectorspace F" ..
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  note FE = \<open>F \<unlhd> E\<close>
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  {
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    fix c assume cM: "c \<in> chains M" and ex: "\<exists>x. x \<in> c"
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    have "\<Union>c \<in> M"
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      -- \<open>Show that every non-empty chain \<open>c\<close> of \<open>M\<close> has an upper bound in \<open>M\<close>:\<close>
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      -- \<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>
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      unfolding M_def
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    proof (rule norm_pres_extensionI)
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      let ?H = "domain (\<Union>c)"
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      let ?h = "funct (\<Union>c)"
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      have a: "graph ?H ?h = \<Union>c"
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      proof (rule graph_domain_funct)
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        fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"
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        with M_def cM show "z = y" by (rule sup_definite)
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      qed
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      moreover from M cM a have "linearform ?H ?h"
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        by (rule sup_lf)
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      moreover from a M cM ex FE E have "?H \<unlhd> E"
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        by (rule sup_subE)
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      moreover from a M cM ex FE have "F \<unlhd> ?H"
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        by (rule sup_supF)
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      moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
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        by (rule sup_ext)
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      moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"
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        by (rule sup_norm_pres)
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      ultimately show "\<exists>H h. \<Union>c = graph H h
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          \<and> linearform H h
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          \<and> H \<unlhd> E
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          \<and> F \<unlhd> H
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          \<and> graph F f \<subseteq> graph H h
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          \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast
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    qed
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  }
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  then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"
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  -- \<open>With Zorn's Lemma we can conclude that there is a maximal element in \<open>M\<close>. \<^smallskip>\<close>
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  proof (rule Zorn's_Lemma)
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      -- \<open>We show that \<open>M\<close> is non-empty:\<close>
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    show "graph F f \<in> M"
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      unfolding M_def
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    proof (rule norm_pres_extensionI2)
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      show "linearform F f" by fact
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      show "F \<unlhd> E" by fact
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      from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)
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      show "graph F f \<subseteq> graph F f" ..
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      show "\<forall>x\<in>F. f x \<le> p x" by fact
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    qed
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  qed
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  then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
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    by blast
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  from gM obtain H h where
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      g_rep: "g = graph H h"
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    and linearform: "linearform H h"
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    and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"
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    and graphs: "graph F f \<subseteq> graph H h"
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    and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..
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      -- \<open>\<open>g\<close> is a norm-preserving extension of \<open>f\<close>, in other words:\<close>
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      -- \<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>
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      -- \<open>and \<open>h\<close> is an extension of \<open>f\<close> that is again bounded by \<open>p\<close>. \<^smallskip>\<close>
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  from HE E have H: "vectorspace H"
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    by (rule subspace.vectorspace)
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  have HE_eq: "H = E"
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    -- \<open>We show that \<open>h\<close> is defined on whole \<open>E\<close> by classical contradiction. \<^smallskip>\<close>
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  proof (rule classical)
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    assume neq: "H \<noteq> E"
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      -- \<open>Assume \<open>h\<close> is not defined on whole \<open>E\<close>. Then show that \<open>h\<close> can be extended\<close>
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      -- \<open>in a norm-preserving way to a function \<open>h'\<close> with the graph \<open>g'\<close>. \<^smallskip>\<close>
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    have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
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    proof -
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      from HE have "H \<subseteq> E" ..
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      with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast
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      obtain x': "x' \<noteq> 0"
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      proof
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        show "x' \<noteq> 0"
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        proof
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          assume "x' = 0"
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parents: 12018
diff changeset
   140
          with H have "x' \<in> H" by (simp only: vectorspace.zero)
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   141
          with \<open>x' \<notin> H\<close> show False by contradiction
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   142
        qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   143
      qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   144
47445
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 46867
diff changeset
   145
      def H' \<equiv> "H + lin x'"
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   146
        -- \<open>Define \<open>H'\<close> as the direct sum of \<open>H\<close> and the linear closure of \<open>x'\<close>. \<^smallskip>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   147
      have HH': "H \<unlhd> H'"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   148
      proof (unfold H'_def)
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   149
        from x'E have "vectorspace (lin x')" ..
47445
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 46867
diff changeset
   150
        with H show "H \<unlhd> H + lin x'" ..
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   151
      qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   152
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   153
      obtain xi where
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   154
        xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   155
          \<and> xi \<le> p (y + x') - h y"
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   156
        -- \<open>Pick a real number \<open>\<xi>\<close> that fulfills certain inequality; this will\<close>
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   157
        -- \<open>be used to establish that \<open>h'\<close> is a norm-preserving extension of \<open>h\<close>.
61486
3590367b0ce9 tuned document;
wenzelm
parents: 59197
diff changeset
   158
           \label{ex-xi-use}\<^smallskip>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   159
      proof -
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   160
        from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   161
            \<and> xi \<le> p (y + x') - h y"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   162
        proof (rule ex_xi)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   163
          fix u v assume u: "u \<in> H" and v: "v \<in> H"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   164
          with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   165
          from H u v linearform have "h v - h u = h (v - u)"
13547
wenzelm
parents: 13515
diff changeset
   166
            by (simp add: linearform.diff)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   167
          also from hp and H u v have "\<dots> \<le> p (v - u)"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   168
            by (simp only: vectorspace.diff_closed)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   169
          also from x'E uE vE have "v - u = x' + - x' + v + - u"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   170
            by (simp add: diff_eq1)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   171
          also from x'E uE vE have "\<dots> = v + x' + - (u + x')"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   172
            by (simp add: add_ac)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   173
          also from x'E uE vE have "\<dots> = (v + x') - (u + x')"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   174
            by (simp add: diff_eq1)
13547
wenzelm
parents: 13515
diff changeset
   175
          also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   176
            by (simp add: diff_subadditive)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   177
          finally have "h v - h u \<le> p (v + x') + p (u + x')" .
13547
wenzelm
parents: 13515
diff changeset
   178
          then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   179
        qed
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   180
        then show thesis by (blast intro: that)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   181
      qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   182
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   183
      def h' \<equiv> "\<lambda>x. let (y, a) =
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   184
          SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   185
        -- \<open>Define the extension \<open>h'\<close> of \<open>h\<close> to \<open>H'\<close> using \<open>\<xi>\<close>. \<^smallskip>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   186
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   187
      have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   188
        -- \<open>\<open>h'\<close> is an extension of \<open>h\<close> \dots \<^smallskip>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   189
      proof
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   190
        show "g \<subseteq> graph H' h'"
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   191
        proof -
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   192
          have  "graph H h \<subseteq> graph H' h'"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   193
          proof (rule graph_extI)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   194
            fix t assume t: "t \<in> H"
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   195
            from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   196
              using \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> by (rule decomp_H'_H)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   197
            with h'_def show "h t = h' t" by (simp add: Let_def)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   198
          next
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   199
            from HH' show "H \<subseteq> H'" ..
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   200
          qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   201
          with g_rep show ?thesis by (simp only:)
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   202
        qed
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   203
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   204
        show "g \<noteq> graph H' h'"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   205
        proof -
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   206
          have "graph H h \<noteq> graph H' h'"
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   207
          proof
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   208
            assume eq: "graph H h = graph H' h'"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   209
            have "x' \<in> H'"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   210
              unfolding H'_def
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   211
            proof
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   212
              from H show "0 \<in> H" by (rule vectorspace.zero)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   213
              from x'E show "x' \<in> lin x'" by (rule x_lin_x)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   214
              from x'E show "x' = 0 + x'" by simp
9261
879e0f0cd047 removed sorry;
bauerg
parents: 9256
diff changeset
   215
            qed
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   216
            then have "(x', h' x') \<in> graph H' h'" ..
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   217
            with eq have "(x', h' x') \<in> graph H h" by (simp only:)
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   218
            then have "x' \<in> H" ..
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   219
            with \<open>x' \<notin> H\<close> show False by contradiction
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   220
          qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   221
          with g_rep show ?thesis by simp
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   222
        qed
9256
f9a6670427fb completed TYPES version of HahnBanach;
bauerg
parents: 9035
diff changeset
   223
      qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   224
      moreover have "graph H' h' \<in> M"
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   225
        -- \<open>and \<open>h'\<close> is norm-preserving. \<^smallskip>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   226
      proof (unfold M_def)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   227
        show "graph H' h' \<in> norm_pres_extensions E p F f"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   228
        proof (rule norm_pres_extensionI2)
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   229
          show "linearform H' h'"
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   230
            using h'_def H'_def HE linearform \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   231
            by (rule h'_lf)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   232
          show "H' \<unlhd> E"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   233
          unfolding H'_def
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   234
          proof
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   235
            show "H \<unlhd> E" by fact
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   236
            show "vectorspace E" by fact
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   237
            from x'E show "lin x' \<unlhd> E" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   238
          qed
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   239
          from H \<open>F \<unlhd> H\<close> HH' show FH': "F \<unlhd> H'"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   240
            by (rule vectorspace.subspace_trans)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   241
          show "graph F f \<subseteq> graph H' h'"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   242
          proof (rule graph_extI)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   243
            fix x assume x: "x \<in> F"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   244
            with graphs have "f x = h x" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   245
            also have "\<dots> = h x + 0 * xi" by simp
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   246
            also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   247
              by (simp add: Let_def)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   248
            also have "(x, 0) =
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   249
                (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   250
              using E HE
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   251
            proof (rule decomp_H'_H [symmetric])
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   252
              from FH x show "x \<in> H" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   253
              from x' show "x' \<noteq> 0" .
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   254
              show "x' \<notin> H" by fact
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   255
              show "x' \<in> E" by fact
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   256
            qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   257
            also have
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   258
              "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   259
              in h y + a * xi) = h' x" by (simp only: h'_def)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   260
            finally show "f x = h' x" .
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   261
          next
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   262
            from FH' show "F \<subseteq> H'" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   263
          qed
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   264
          show "\<forall>x \<in> H'. h' x \<le> p x"
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   265
            using h'_def H'_def \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E HE
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   266
              \<open>seminorm E p\<close> linearform and hp xi
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   267
            by (rule h'_norm_pres)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   268
        qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   269
      qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   270
      ultimately show ?thesis ..
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   271
    qed
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   272
    then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   273
      -- \<open>So the graph \<open>g\<close> of \<open>h\<close> cannot be maximal. Contradiction! \<^smallskip>\<close>
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   274
    with gx show "H = E" by contradiction
9035
371f023d3dbd removed explicit terminator (";");
wenzelm
parents: 9013
diff changeset
   275
  qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   276
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   277
  from HE_eq and linearform have "linearform E h"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   278
    by (simp only:)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   279
  moreover have "\<forall>x \<in> F. h x = f x"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   280
  proof
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   281
    fix x assume "x \<in> F"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   282
    with graphs have "f x = h x" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   283
    then show "h x = f x" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   284
  qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   285
  moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   286
    by (simp only:)
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   287
  ultimately show ?thesis by blast
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   288
qed
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   289
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   290
59197
c112a24446d4 tuned whitespace;
wenzelm
parents: 58889
diff changeset
   291
subsection \<open>Alternative formulation\<close>
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   292
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   293
text \<open>
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   294
  The following alternative formulation of the Hahn-Banach
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   295
  Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear form
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   296
  \<open>f\<close> and a seminorm \<open>p\<close> the following inequality are
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   297
  equivalent:\footnote{This was shown in lemma @{thm [source] abs_ineq_iff}
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   298
  (see page \pageref{abs-ineq-iff}).}
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   299
  \begin{center}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   300
  \begin{tabular}{lll}
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   301
  \<open>\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x\<close> & and &
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   302
  \<open>\<forall>x \<in> H. h x \<le> p x\<close> \\
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   303
  \end{tabular}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   304
  \end{center}
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   305
\<close>
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   306
31795
be3e1cc5005c standard naming conventions for session and theories;
wenzelm
parents: 29291
diff changeset
   307
theorem abs_Hahn_Banach:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   308
  assumes E: "vectorspace E" and FE: "subspace F E"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   309
    and lf: "linearform F f" and sn: "seminorm E p"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   310
  assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   311
  shows "\<exists>g. linearform E g
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   312
    \<and> (\<forall>x \<in> F. g x = f x)
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   313
    \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   314
proof -
29234
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   315
  interpret vectorspace E by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   316
  interpret subspace F E by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   317
  interpret linearform F f by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   318
  interpret seminorm E p by fact
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   319
  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)"
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   320
    using E FE sn lf
31795
be3e1cc5005c standard naming conventions for session and theories;
wenzelm
parents: 29291
diff changeset
   321
  proof (rule Hahn_Banach)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   322
    show "\<forall>x \<in> F. f x \<le> p x"
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   323
      using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   324
  qed
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   325
  then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   326
      and **: "\<forall>x \<in> E. g x \<le> p x" by blast
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   327
  have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   328
    using _ E sn lg **
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   329
  proof (rule abs_ineq_iff [THEN iffD2])
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   330
    show "E \<unlhd> E" ..
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   331
  qed
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   332
  with lg * show ?thesis by blast
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   333
qed
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   334
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   335
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   336
subsection \<open>The Hahn-Banach Theorem for normed spaces\<close>
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   337
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   338
text \<open>
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   339
  Every continuous linear form \<open>f\<close> on a subspace \<open>F\<close> of a norm space \<open>E\<close>,
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   340
  can be extended to a continuous linear form \<open>g\<close> on \<open>E\<close> such that \<open>\<parallel>f\<parallel> =
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   341
  \<parallel>g\<parallel>\<close>.
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   342
\<close>
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   343
31795
be3e1cc5005c standard naming conventions for session and theories;
wenzelm
parents: 29291
diff changeset
   344
theorem norm_Hahn_Banach:
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   345
  fixes V and norm ("\<parallel>_\<parallel>")
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   346
  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}"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   347
  fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   348
  defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   349
  assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"
46867
0883804b67bb modernized locale expression, with some fluctuation of arguments;
wenzelm
parents: 44190
diff changeset
   350
    and linearform: "linearform F f" and "continuous F f norm"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   351
  shows "\<exists>g. linearform E g
46867
0883804b67bb modernized locale expression, with some fluctuation of arguments;
wenzelm
parents: 44190
diff changeset
   352
     \<and> continuous E g norm
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   353
     \<and> (\<forall>x \<in> F. g x = f x)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   354
     \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   355
proof -
29234
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   356
  interpret normed_vectorspace E norm by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   357
  interpret normed_vectorspace_with_fn_norm E norm B fn_norm
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   358
    by (auto simp: B_def fn_norm_def) intro_locales
29234
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   359
  interpret subspace F E by fact
60f7fb56f8cd More porting to new locales.
ballarin
parents: 28823
diff changeset
   360
  interpret linearform F f by fact
46867
0883804b67bb modernized locale expression, with some fluctuation of arguments;
wenzelm
parents: 44190
diff changeset
   361
  interpret continuous F f norm by fact
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27612
diff changeset
   362
  have E: "vectorspace E" by intro_locales
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 27612
diff changeset
   363
  have F: "vectorspace F" by rule intro_locales
14214
5369d671f100 Improvements to Isar/Locales: premises generated by "includes" elements
ballarin
parents: 13547
diff changeset
   364
  have F_norm: "normed_vectorspace F norm"
23378
1d138d6bb461 tuned proofs: avoid implicit prems;
wenzelm
parents: 19984
diff changeset
   365
    using FE E_norm by (rule subspace_normed_vs)
13547
wenzelm
parents: 13515
diff changeset
   366
  have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   367
    by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   368
      [OF normed_vectorspace_with_fn_norm.intro,
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   369
       OF F_norm \<open>continuous F f norm\<close> , folded B_def fn_norm_def])
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   370
  txt \<open>We define a function \<open>p\<close> on \<open>E\<close> as follows:
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   371
    \<open>p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   372
  def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   373
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   374
  txt \<open>\<open>p\<close> is a seminorm on \<open>E\<close>:\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   375
  have q: "seminorm E p"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   376
  proof
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   377
    fix x y a assume x: "x \<in> E" and y: "y \<in> E"
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   378
    
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   379
    txt \<open>\<open>p\<close> is positive definite:\<close>
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   380
    have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   381
    moreover from x have "0 \<le> \<parallel>x\<parallel>" ..
14710
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   382
    ultimately show "0 \<le> p x"  
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   383
      by (simp add: p_def zero_le_mult_iff)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   384
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   385
    txt \<open>\<open>p\<close> is absolutely homogeneous:\<close>
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   386
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   387
    show "p (a \<cdot> x) = \<bar>a\<bar> * p x"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   388
    proof -
13547
wenzelm
parents: 13515
diff changeset
   389
      have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)
wenzelm
parents: 13515
diff changeset
   390
      also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)
wenzelm
parents: 13515
diff changeset
   391
      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
wenzelm
parents: 13515
diff changeset
   392
      also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   393
      finally show ?thesis .
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   394
    qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   395
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   396
    txt \<open>Furthermore, \<open>p\<close> is subadditive:\<close>
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   397
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   398
    show "p (x + y) \<le> p x + p y"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   399
    proof -
13547
wenzelm
parents: 13515
diff changeset
   400
      have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)
14710
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   401
      also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   402
      from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   403
      with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
247615bfffb8 replaced Aux.thy by RealLemmas.thy
bauerg
parents: 14334
diff changeset
   404
        by (simp add: mult_left_mono)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47445
diff changeset
   405
      also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left)
13547
wenzelm
parents: 13515
diff changeset
   406
      also have "\<dots> = p x + p y" by (simp only: p_def)
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   407
      finally show ?thesis .
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   408
    qed
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   409
  qed
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   410
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   411
  txt \<open>\<open>f\<close> is bounded by \<open>p\<close>.\<close>
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   412
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   413
  have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   414
  proof
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   415
    fix x assume "x \<in> F"
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   416
    with \<open>continuous F f norm\<close> and linearform
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   417
    show "\<bar>f x\<bar> \<le> p x"
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   418
      unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   419
        [OF normed_vectorspace_with_fn_norm.intro,
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   420
         OF F_norm, folded B_def fn_norm_def])
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   421
  qed
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   422
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   423
  txt \<open>Using the fact that \<open>p\<close> is a seminorm and \<open>f\<close> is bounded by \<open>p\<close> we
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   424
    can apply the Hahn-Banach Theorem for real vector spaces. So \<open>f\<close> can be
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   425
    extended in a norm-preserving way to some function \<open>g\<close> on the whole
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   426
    vector space \<open>E\<close>.\<close>
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   427
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   428
  with E FE linearform q obtain g where
27612
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   429
      linearformE: "linearform E g"
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   430
    and a: "\<forall>x \<in> F. g x = f x"
d3eb431db035 modernized specifications and proofs;
wenzelm
parents: 27611
diff changeset
   431
    and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
31795
be3e1cc5005c standard naming conventions for session and theories;
wenzelm
parents: 29291
diff changeset
   432
    by (rule abs_Hahn_Banach [elim_format]) iprover
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   433
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   434
  txt \<open>We furthermore have to show that \<open>g\<close> is also continuous:\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   435
46867
0883804b67bb modernized locale expression, with some fluctuation of arguments;
wenzelm
parents: 44190
diff changeset
   436
  have g_cont: "continuous E g norm" using linearformE
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   437
  proof
9503
wenzelm
parents: 9475
diff changeset
   438
    fix x assume "x \<in> E"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   439
    with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   440
      by (simp only: p_def)
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   441
  qed
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   442
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   443
  txt \<open>To complete the proof, we show that \<open>\<parallel>g\<parallel> = \<parallel>f\<parallel>\<close>.\<close>
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   444
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   445
  have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   446
  proof (rule order_antisym)
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   447
    txt \<open>
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   448
      First we show \<open>\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>\<close>. The function norm \<open>\<parallel>g\<parallel>\<close> is defined as the
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   449
      smallest \<open>c \<in> \<real>\<close> such that
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   450
      \begin{center}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   451
      \begin{tabular}{l}
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   452
      \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>\<close>
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   453
      \end{tabular}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   454
      \end{center}
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   455
      \<^noindent> Furthermore holds
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   456
      \begin{center}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   457
      \begin{tabular}{l}
61539
a29295dac1ca isabelle update_cartouches -t;
wenzelm
parents: 61486
diff changeset
   458
      \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   459
      \end{tabular}
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   460
      \end{center}
61540
f92bf6674699 tuned document;
wenzelm
parents: 61539
diff changeset
   461
    \<close>
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   462
50918
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   463
    from g_cont _ ge_zero
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   464
    show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   465
    proof
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   466
      fix x assume "x \<in> E"
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   467
      with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   468
        by (simp only: p_def)
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   469
    qed
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   470
58744
c434e37f290e update_cartouches;
wenzelm
parents: 58622
diff changeset
   471
    txt \<open>The other direction is achieved by a similar argument.\<close>
13515
a6a7025fd7e8 updated to use locales (still some rough edges);
wenzelm
parents: 12018
diff changeset
   472
13547
wenzelm
parents: 13515
diff changeset
   473
    show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
27611
2c01c0bdb385 Removed uses of context element includes.
ballarin
parents: 23378
diff changeset
   474
    proof (rule normed_vectorspace_with_fn_norm.fn_norm_least
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   475
        [OF normed_vectorspace_with_fn_norm.intro,
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   476
         OF F_norm, folded B_def fn_norm_def])
50918
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   477
      fix x assume x: "x \<in> F"
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   478
      show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   479
      proof -
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   480
        from a x have "g x = f x" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   481
        then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   482
        also from g_cont
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   483
        have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   484
        proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   485
          from FE x show "x \<in> E" ..
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   486
        qed
50918
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   487
        finally show ?thesis .
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   488
      qed
50918
3b6417e9f73e tuned proofs;
wenzelm
parents: 49962
diff changeset
   489
    next
13547
wenzelm
parents: 13515
diff changeset
   490
      show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   491
        using g_cont
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 31795
diff changeset
   492
        by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
46867
0883804b67bb modernized locale expression, with some fluctuation of arguments;
wenzelm
parents: 44190
diff changeset
   493
      show "continuous F f norm" by fact
10687
c186279eecea tuned HOL/Real/HahnBanach;
wenzelm
parents: 10387
diff changeset
   494
    qed
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   495
  qed
13547
wenzelm
parents: 13515
diff changeset
   496
  with linearformE a g_cont show ?thesis by blast
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   497
qed
9374
153853af318b - xsymbols for
bauerg
parents: 9261
diff changeset
   498
9475
b24516d96847 adapted obtain;
wenzelm
parents: 9394
diff changeset
   499
end