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src/HOL/Hahn_Banach/Hahn_Banach.thy

author | wenzelm |

Mon, 02 Nov 2015 16:02:32 +0100 | |

changeset 61543 | de44d4fa5ef0 |

parent 61540 | f92bf6674699 |

child 61583 | c2b7033fa0ba |

permissions | -rw-r--r-- |

tuned whitespace;

(* Title: HOL/Hahn_Banach/Hahn_Banach.thy Author: Gertrud Bauer, TU Munich *) section \<open>The Hahn-Banach Theorem\<close> theory Hahn_Banach imports Hahn_Banach_Lemmas begin text \<open> We present the proof of two different versions of the Hahn-Banach Theorem, closely following @{cite \<open>\S36\<close> "Heuser:1986"}. \<close> subsection \<open>The Hahn-Banach Theorem for vector spaces\<close> paragraph \<open>Hahn-Banach Theorem.\<close> text \<open> Let \<open>F\<close> be a subspace of a real vector space \<open>E\<close>, let \<open>p\<close> be a semi-norm on \<open>E\<close>, and \<open>f\<close> be a linear form defined on \<open>F\<close> such that \<open>f\<close> is bounded 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 form \<open>h\<close> on \<open>E\<close> such that \<open>h\<close> is norm-preserving, i.e. \<open>h\<close> is also bounded by \<open>p\<close>. \<close> paragraph \<open>Proof Sketch.\<close> text \<open> \<^enum> Define \<open>M\<close> as the set of norm-preserving extensions of \<open>f\<close> to subspaces of \<open>E\<close>. The linear forms in \<open>M\<close> are ordered by domain extension. \<^enum> We show that every non-empty chain in \<open>M\<close> has an upper bound in \<open>M\<close>. \<^enum> With Zorn's Lemma we conclude that there is a maximal function \<open>g\<close> in \<open>M\<close>. \<^enum> The domain \<open>H\<close> of \<open>g\<close> is the whole space \<open>E\<close>, as shown by classical contradiction: \<^item> Assuming \<open>g\<close> is not defined on whole \<open>E\<close>, it can still be extended in a norm-preserving way to a super-space \<open>H'\<close> of \<open>H\<close>. \<^item> Thus \<open>g\<close> can not be maximal. Contradiction! \<close> theorem Hahn_Banach: assumes E: "vectorspace E" and "subspace F E" and "seminorm E p" and "linearform F f" assumes fp: "\<forall>x \<in> F. f x \<le> p x" shows "\<exists>h. linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)" -- \<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> -- \<open>and \<open>f\<close> a linear form on \<open>F\<close> such that \<open>f\<close> is bounded by \<open>p\<close>,\<close> -- \<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> proof - interpret vectorspace E by fact interpret subspace F E by fact interpret seminorm E p by fact interpret linearform F f by fact def M \<equiv> "norm_pres_extensions E p F f" then have M: "M = \<dots>" by (simp only:) from E have F: "vectorspace F" .. note FE = \<open>F \<unlhd> E\<close> { fix c assume cM: "c \<in> chains M" and ex: "\<exists>x. x \<in> c" have "\<Union>c \<in> M" -- \<open>Show that every non-empty chain \<open>c\<close> of \<open>M\<close> has an upper bound in \<open>M\<close>:\<close> -- \<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> unfolding M_def proof (rule norm_pres_extensionI) let ?H = "domain (\<Union>c)" let ?h = "funct (\<Union>c)" have a: "graph ?H ?h = \<Union>c" proof (rule graph_domain_funct) fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c" with M_def cM show "z = y" by (rule sup_definite) qed moreover from M cM a have "linearform ?H ?h" by (rule sup_lf) moreover from a M cM ex FE E have "?H \<unlhd> E" by (rule sup_subE) moreover from a M cM ex FE have "F \<unlhd> ?H" by (rule sup_supF) moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h" by (rule sup_ext) moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x" by (rule sup_norm_pres) ultimately show "\<exists>H h. \<Union>c = graph H h \<and> linearform H h \<and> H \<unlhd> E \<and> F \<unlhd> H \<and> graph F f \<subseteq> graph H h \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast qed } then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g" -- \<open>With Zorn's Lemma we can conclude that there is a maximal element in \<open>M\<close>. \<^smallskip>\<close> proof (rule Zorn's_Lemma) -- \<open>We show that \<open>M\<close> is non-empty:\<close> show "graph F f \<in> M" unfolding M_def proof (rule norm_pres_extensionI2) show "linearform F f" by fact show "F \<unlhd> E" by fact from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl) show "graph F f \<subseteq> graph F f" .. show "\<forall>x\<in>F. f x \<le> p x" by fact qed qed then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x" by blast from gM obtain H h where g_rep: "g = graph H h" and linearform: "linearform H h" and HE: "H \<unlhd> E" and FH: "F \<unlhd> H" and graphs: "graph F f \<subseteq> graph H h" and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def .. -- \<open>\<open>g\<close> is a norm-preserving extension of \<open>f\<close>, in other words:\<close> -- \<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> -- \<open>and \<open>h\<close> is an extension of \<open>f\<close> that is again bounded by \<open>p\<close>. \<^smallskip>\<close> from HE E have H: "vectorspace H" by (rule subspace.vectorspace) have HE_eq: "H = E" -- \<open>We show that \<open>h\<close> is defined on whole \<open>E\<close> by classical contradiction. \<^smallskip>\<close> proof (rule classical) assume neq: "H \<noteq> E" -- \<open>Assume \<open>h\<close> is not defined on whole \<open>E\<close>. Then show that \<open>h\<close> can be extended\<close> -- \<open>in a norm-preserving way to a function \<open>h'\<close> with the graph \<open>g'\<close>. \<^smallskip>\<close> have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'" proof - from HE have "H \<subseteq> E" .. with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast obtain x': "x' \<noteq> 0" proof show "x' \<noteq> 0" proof assume "x' = 0" with H have "x' \<in> H" by (simp only: vectorspace.zero) with \<open>x' \<notin> H\<close> show False by contradiction qed qed def H' \<equiv> "H + lin x'" -- \<open>Define \<open>H'\<close> as the direct sum of \<open>H\<close> and the linear closure of \<open>x'\<close>. \<^smallskip>\<close> have HH': "H \<unlhd> H'" proof (unfold H'_def) from x'E have "vectorspace (lin x')" .. with H show "H \<unlhd> H + lin x'" .. qed obtain xi where xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi \<and> xi \<le> p (y + x') - h y" -- \<open>Pick a real number \<open>\<xi>\<close> that fulfills certain inequality; this will\<close> -- \<open>be used to establish that \<open>h'\<close> is a norm-preserving extension of \<open>h\<close>. \label{ex-xi-use}\<^smallskip>\<close> proof - from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi \<and> xi \<le> p (y + x') - h y" proof (rule ex_xi) fix u v assume u: "u \<in> H" and v: "v \<in> H" with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto from H u v linearform have "h v - h u = h (v - u)" by (simp add: linearform.diff) also from hp and H u v have "\<dots> \<le> p (v - u)" by (simp only: vectorspace.diff_closed) also from x'E uE vE have "v - u = x' + - x' + v + - u" by (simp add: diff_eq1) also from x'E uE vE have "\<dots> = v + x' + - (u + x')" by (simp add: add_ac) also from x'E uE vE have "\<dots> = (v + x') - (u + x')" by (simp add: diff_eq1) also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')" by (simp add: diff_subadditive) finally have "h v - h u \<le> p (v + x') + p (u + x')" . then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp qed then show thesis by (blast intro: that) qed def h' \<equiv> "\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi" -- \<open>Define the extension \<open>h'\<close> of \<open>h\<close> to \<open>H'\<close> using \<open>\<xi>\<close>. \<^smallskip>\<close> have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'" -- \<open>\<open>h'\<close> is an extension of \<open>h\<close> \dots \<^smallskip>\<close> proof show "g \<subseteq> graph H' h'" proof - have "graph H h \<subseteq> graph H' h'" proof (rule graph_extI) fix t assume t: "t \<in> H" from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)" using \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> by (rule decomp_H'_H) with h'_def show "h t = h' t" by (simp add: Let_def) next from HH' show "H \<subseteq> H'" .. qed with g_rep show ?thesis by (simp only:) qed show "g \<noteq> graph H' h'" proof - have "graph H h \<noteq> graph H' h'" proof assume eq: "graph H h = graph H' h'" have "x' \<in> H'" unfolding H'_def proof from H show "0 \<in> H" by (rule vectorspace.zero) from x'E show "x' \<in> lin x'" by (rule x_lin_x) from x'E show "x' = 0 + x'" by simp qed then have "(x', h' x') \<in> graph H' h'" .. with eq have "(x', h' x') \<in> graph H h" by (simp only:) then have "x' \<in> H" .. with \<open>x' \<notin> H\<close> show False by contradiction qed with g_rep show ?thesis by simp qed qed moreover have "graph H' h' \<in> M" -- \<open>and \<open>h'\<close> is norm-preserving. \<^smallskip>\<close> proof (unfold M_def) show "graph H' h' \<in> norm_pres_extensions E p F f" proof (rule norm_pres_extensionI2) show "linearform H' h'" using h'_def H'_def HE linearform \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E by (rule h'_lf) show "H' \<unlhd> E" unfolding H'_def proof show "H \<unlhd> E" by fact show "vectorspace E" by fact from x'E show "lin x' \<unlhd> E" .. qed from H \<open>F \<unlhd> H\<close> HH' show FH': "F \<unlhd> H'" by (rule vectorspace.subspace_trans) show "graph F f \<subseteq> graph H' h'" proof (rule graph_extI) fix x assume x: "x \<in> F" with graphs have "f x = h x" .. also have "\<dots> = h x + 0 * xi" by simp also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)" by (simp add: Let_def) also have "(x, 0) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)" using E HE proof (rule decomp_H'_H [symmetric]) from FH x show "x \<in> H" .. from x' show "x' \<noteq> 0" . show "x' \<notin> H" by fact show "x' \<in> E" by fact qed also have "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) in h y + a * xi) = h' x" by (simp only: h'_def) finally show "f x = h' x" . next from FH' show "F \<subseteq> H'" .. qed show "\<forall>x \<in> H'. h' x \<le> p x" using h'_def H'_def \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E HE \<open>seminorm E p\<close> linearform and hp xi by (rule h'_norm_pres) qed qed ultimately show ?thesis .. qed then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp -- \<open>So the graph \<open>g\<close> of \<open>h\<close> cannot be maximal. Contradiction! \<^smallskip>\<close> with gx show "H = E" by contradiction qed from HE_eq and linearform have "linearform E h" by (simp only:) moreover have "\<forall>x \<in> F. h x = f x" proof fix x assume "x \<in> F" with graphs have "f x = h x" .. then show "h x = f x" .. qed moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x" by (simp only:) ultimately show ?thesis by blast qed subsection \<open>Alternative formulation\<close> text \<open> The following alternative formulation of the Hahn-Banach Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear form \<open>f\<close> and a seminorm \<open>p\<close> the following inequality are equivalent:\footnote{This was shown in lemma @{thm [source] abs_ineq_iff} (see page \pageref{abs-ineq-iff}).} \begin{center} \begin{tabular}{lll} \<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> \\ \end{tabular} \end{center} \<close> theorem abs_Hahn_Banach: assumes E: "vectorspace E" and FE: "subspace F E" and lf: "linearform F f" and sn: "seminorm E p" assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x" shows "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x) \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)" proof - interpret vectorspace E by fact interpret subspace F E by fact interpret linearform F f by fact interpret seminorm E p by fact have "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x) \<and> (\<forall>x \<in> E. g x \<le> p x)" using E FE sn lf proof (rule Hahn_Banach) show "\<forall>x \<in> F. f x \<le> p x" using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1]) qed then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x" and **: "\<forall>x \<in> E. g x \<le> p x" by blast have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x" using _ E sn lg ** proof (rule abs_ineq_iff [THEN iffD2]) show "E \<unlhd> E" .. qed with lg * show ?thesis by blast qed subsection \<open>The Hahn-Banach Theorem for normed spaces\<close> text \<open> Every continuous linear form \<open>f\<close> on a subspace \<open>F\<close> of a norm space \<open>E\<close>, can be extended to a continuous linear form \<open>g\<close> on \<open>E\<close> such that \<open>\<parallel>f\<parallel> = \<parallel>g\<parallel>\<close>. \<close> theorem norm_Hahn_Banach: fixes V and norm ("\<parallel>_\<parallel>") fixes B defines "\<And>V f. B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}" fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999) defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)" assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E" and linearform: "linearform F f" and "continuous F f norm" shows "\<exists>g. linearform E g \<and> continuous E g norm \<and> (\<forall>x \<in> F. g x = f x) \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F" proof - interpret normed_vectorspace E norm by fact interpret normed_vectorspace_with_fn_norm E norm B fn_norm by (auto simp: B_def fn_norm_def) intro_locales interpret subspace F E by fact interpret linearform F f by fact interpret continuous F f norm by fact have E: "vectorspace E" by intro_locales have F: "vectorspace F" by rule intro_locales have F_norm: "normed_vectorspace F norm" using FE E_norm by (rule subspace_normed_vs) have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero [OF normed_vectorspace_with_fn_norm.intro, OF F_norm \<open>continuous F f norm\<close> , folded B_def fn_norm_def]) txt \<open>We define a function \<open>p\<close> on \<open>E\<close> as follows: \<open>p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>\<close> def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>" txt \<open>\<open>p\<close> is a seminorm on \<open>E\<close>:\<close> have q: "seminorm E p" proof fix x y a assume x: "x \<in> E" and y: "y \<in> E" txt \<open>\<open>p\<close> is positive definite:\<close> have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero) moreover from x have "0 \<le> \<parallel>x\<parallel>" .. ultimately show "0 \<le> p x" by (simp add: p_def zero_le_mult_iff) txt \<open>\<open>p\<close> is absolutely homogeneous:\<close> show "p (a \<cdot> x) = \<bar>a\<bar> * p x" proof - have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def) also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous) also have "\<parallel>f\<parallel>\<hyphen>F * (\<bar>a\<bar> * \<parallel>x\<parallel>) = \<bar>a\<bar> * (\<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>)" by simp also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def) finally show ?thesis . qed txt \<open>Furthermore, \<open>p\<close> is subadditive:\<close> show "p (x + y) \<le> p x + p y" proof - have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def) also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero) from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" .. with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)" by (simp add: mult_left_mono) also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left) also have "\<dots> = p x + p y" by (simp only: p_def) finally show ?thesis . qed qed txt \<open>\<open>f\<close> is bounded by \<open>p\<close>.\<close> have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x" proof fix x assume "x \<in> F" with \<open>continuous F f norm\<close> and linearform show "\<bar>f x\<bar> \<le> p x" unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong [OF normed_vectorspace_with_fn_norm.intro, OF F_norm, folded B_def fn_norm_def]) qed txt \<open>Using the fact that \<open>p\<close> is a seminorm and \<open>f\<close> is bounded by \<open>p\<close> we can apply the Hahn-Banach Theorem for real vector spaces. So \<open>f\<close> can be extended in a norm-preserving way to some function \<open>g\<close> on the whole vector space \<open>E\<close>.\<close> with E FE linearform q obtain g where linearformE: "linearform E g" and a: "\<forall>x \<in> F. g x = f x" and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x" by (rule abs_Hahn_Banach [elim_format]) iprover txt \<open>We furthermore have to show that \<open>g\<close> is also continuous:\<close> have g_cont: "continuous E g norm" using linearformE proof fix x assume "x \<in> E" with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>" by (simp only: p_def) qed txt \<open>To complete the proof, we show that \<open>\<parallel>g\<parallel> = \<parallel>f\<parallel>\<close>.\<close> have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F" proof (rule order_antisym) txt \<open> First we show \<open>\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>\<close>. The function norm \<open>\<parallel>g\<parallel>\<close> is defined as the smallest \<open>c \<in> \<real>\<close> such that \begin{center} \begin{tabular}{l} \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>\<close> \end{tabular} \end{center} \<^noindent> Furthermore holds \begin{center} \begin{tabular}{l} \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close> \end{tabular} \end{center} \<close> from g_cont _ ge_zero show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F" proof fix x assume "x \<in> E" with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>" by (simp only: p_def) qed txt \<open>The other direction is achieved by a similar argument.\<close> show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E" proof (rule normed_vectorspace_with_fn_norm.fn_norm_least [OF normed_vectorspace_with_fn_norm.intro, OF F_norm, folded B_def fn_norm_def]) fix x assume x: "x \<in> F" show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" proof - from a x have "g x = f x" .. then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:) also from g_cont have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def]) from FE x show "x \<in> E" .. qed finally show ?thesis . qed next show "0 \<le> \<parallel>g\<parallel>\<hyphen>E" using g_cont by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def]) show "continuous F f norm" by fact qed qed with linearformE a g_cont show ?thesis by blast qed end