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src/HOL/Hahn_Banach/Hahn_Banach.thy

author | wenzelm |

Sat, 07 Apr 2012 16:41:59 +0200 | |

changeset 47389 | e8552cba702d |

parent 46867 | 0883804b67bb |

child 47445 | 69e96e5500df |

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

explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;

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