(* Title: HOL/Hahn_Banach/Hahn_Banach.thy Author: Gertrud Bauer, TU Munich *) section ‹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› paragraph ‹Hahn-Banach Theorem.› text ‹ Let ‹F› be a subspace of a real vector space ‹E›, let ‹p› be a semi-norm on ‹E›, and ‹f› be a linear form defined on ‹F› such that ‹f› is bounded by ‹p›, i.e. ‹∀x ∈ F. f x ≤ p x›. Then ‹f› can be extended to a linear form ‹h› on ‹E› such that ‹h› is norm-preserving, i.e. ‹h› is also bounded by ‹p›. › paragraph ‹Proof Sketch.› text ‹ ▸ Define ‹M› as the set of norm-preserving extensions of ‹f› to subspaces of ‹E›. The linear forms in ‹M› are ordered by domain extension. ▸ We show that every non-empty chain in ‹M› has an upper bound in ‹M›. ▸ With Zorn's Lemma we conclude that there is a maximal function ‹g› in ‹M›. ▸ The domain ‹H› of ‹g› is the whole space ‹E›, as shown by classical contradiction: ▪ Assuming ‹g› is not defined on whole ‹E›, it can still be extended in a norm-preserving way to a super-space ‹H'› of ‹H›. ▪ Thus ‹g› can not be maximal. Contradiction! › theorem Hahn_Banach: assumes E: "vectorspace E" and "subspace F E" and "seminorm E p" and "linearform F f" assumes fp: "∀x ∈ F. f x ≤ p x" shows "∃h. linearform E h ∧ (∀x ∈ F. h x = f x) ∧ (∀x ∈ E. h x ≤ p x)" ― ‹Let ‹E› be a vector space, ‹F› a subspace of ‹E›, ‹p› a seminorm on ‹E›,› ― ‹and ‹f› a linear form on ‹F› such that ‹f› is bounded by ‹p›,› ― ‹then ‹f› can be extended to a linear form ‹h› on ‹E› in a norm-preserving way. ┈› proof - interpret vectorspace E by fact interpret subspace F E by fact interpret seminorm E p by fact interpret linearform F f by fact define M where "M = norm_pres_extensions E p F f" then have M: "M = …" by (simp only:) from E have F: "vectorspace F" .. note FE = ‹F ⊴ E› { fix c assume cM: "c ∈ chains M" and ex: "∃x. x ∈ c" have "⋃c ∈ M" ― ‹Show that every non-empty chain ‹c› of ‹M› has an upper bound in ‹M›:› ― ‹‹⋃c› is greater than any element of the chain ‹c›, so it suffices to show ‹⋃c ∈ M›.› unfolding M_def proof (rule norm_pres_extensionI) let ?H = "domain (⋃c)" let ?h = "funct (⋃c)" have a: "graph ?H ?h = ⋃c" proof (rule graph_domain_funct) fix x y z assume "(x, y) ∈ ⋃c" and "(x, z) ∈ ⋃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 ⊴ E" by (rule sup_subE) moreover from a M cM ex FE have "F ⊴ ?H" by (rule sup_supF) moreover from a M cM ex have "graph F f ⊆ graph ?H ?h" by (rule sup_ext) moreover from a M cM have "∀x ∈ ?H. ?h x ≤ p x" by (rule sup_norm_pres) ultimately show "∃H h. ⋃c = graph H h ∧ linearform H h ∧ H ⊴ E ∧ F ⊴ H ∧ graph F f ⊆ graph H h ∧ (∀x ∈ H. h x ≤ p x)" by blast qed } then have "∃g ∈ M. ∀x ∈ M. g ⊆ x ⟶ x = g" ― ‹With Zorn's Lemma we can conclude that there is a maximal element in ‹M›. ┈› proof (rule Zorn's_Lemma) ― ‹We show that ‹M› is non-empty:› show "graph F f ∈ M" unfolding M_def proof (rule norm_pres_extensionI2) show "linearform F f" by fact show "F ⊴ E" by fact from F show "F ⊴ F" by (rule vectorspace.subspace_refl) show "graph F f ⊆ graph F f" .. show "∀x∈F. f x ≤ p x" by fact qed qed then obtain g where gM: "g ∈ M" and gx: "∀x ∈ M. g ⊆ x ⟶ g = x" by blast from gM obtain H h where g_rep: "g = graph H h" and linearform: "linearform H h" and HE: "H ⊴ E" and FH: "F ⊴ H" and graphs: "graph F f ⊆ graph H h" and hp: "∀x ∈ H. h x ≤ p x" unfolding M_def .. ― ‹‹g› is a norm-preserving extension of ‹f›, in other words:› ― ‹‹g› is the graph of some linear form ‹h› defined on a subspace ‹H› of ‹E›,› ― ‹and ‹h› is an extension of ‹f› that is again bounded by ‹p›. ┈› from HE E have H: "vectorspace H" by (rule subspace.vectorspace) have HE_eq: "H = E" ― ‹We show that ‹h› is defined on whole ‹E› by classical contradiction. ┈› proof (rule classical) assume neq: "H ≠ E" ― ‹Assume ‹h› is not defined on whole ‹E›. Then show that ‹h› can be extended› ― ‹in a norm-preserving way to a function ‹h'› with the graph ‹g'›. ┈› have "∃g' ∈ M. g ⊆ g' ∧ g ≠ g'" proof - from HE have "H ⊆ E" .. with neq obtain x' where x'E: "x' ∈ E" and "x' ∉ H" by blast obtain x': "x' ≠ 0" proof show "x' ≠ 0" proof assume "x' = 0" with H have "x' ∈ H" by (simp only: vectorspace.zero) with ‹x' ∉ H› show False by contradiction qed qed define H' where "H' = H + lin x'" ― ‹Define ‹H'› as the direct sum of ‹H› and the linear closure of ‹x'›. ┈› have HH': "H ⊴ H'" proof (unfold H'_def) from x'E have "vectorspace (lin x')" .. with H show "H ⊴ H + lin x'" .. qed obtain xi where xi: "∀y ∈ H. - p (y + x') - h y ≤ xi ∧ xi ≤ p (y + x') - h y" ― ‹Pick a real number ‹ξ› that fulfills certain inequality; this will› ― ‹be used to establish that ‹h'› is a norm-preserving extension of ‹h›. \label{ex-xi-use}┈› proof - from H have "∃xi. ∀y ∈ H. - p (y + x') - h y ≤ xi ∧ xi ≤ p (y + x') - h y" proof (rule ex_xi) fix u v assume u: "u ∈ H" and v: "v ∈ H" with HE have uE: "u ∈ E" and vE: "v ∈ 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 "… ≤ 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 "… = v + x' + - (u + x')" by (simp add: add_ac) also from x'E uE vE have "… = (v + x') - (u + x')" by (simp add: diff_eq1) also from x'E uE vE E have "p … ≤ p (v + x') + p (u + x')" by (simp add: diff_subadditive) finally have "h v - h u ≤ p (v + x') + p (u + x')" . then show "- p (u + x') - h u ≤ p (v + x') - h v" by simp qed then show thesis by (blast intro: that) qed define h' where "h' x = (let (y, a) = SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H in h y + a * xi)" for x ― ‹Define the extension ‹h'› of ‹h› to ‹H'› using ‹ξ›. ┈› have "g ⊆ graph H' h' ∧ g ≠ graph H' h'" ― ‹‹h'› is an extension of ‹h› \dots ┈› proof show "g ⊆ graph H' h'" proof - have "graph H h ⊆ graph H' h'" proof (rule graph_extI) fix t assume t: "t ∈ H" from E HE t have "(SOME (y, a). t = y + a ⋅ x' ∧ y ∈ H) = (t, 0)" using ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› by (rule decomp_H'_H) with h'_def show "h t = h' t" by (simp add: Let_def) next from HH' show "H ⊆ H'" .. qed with g_rep show ?thesis by (simp only:) qed show "g ≠ graph H' h'" proof - have "graph H h ≠ graph H' h'" proof assume eq: "graph H h = graph H' h'" have "x' ∈ H'" unfolding H'_def proof from H show "0 ∈ H" by (rule vectorspace.zero) from x'E show "x' ∈ lin x'" by (rule x_lin_x) from x'E show "x' = 0 + x'" by simp qed then have "(x', h' x') ∈ graph H' h'" .. with eq have "(x', h' x') ∈ graph H h" by (simp only:) then have "x' ∈ H" .. with ‹x' ∉ H› show False by contradiction qed with g_rep show ?thesis by simp qed qed moreover have "graph H' h' ∈ M" ― ‹and ‹h'› is norm-preserving. ┈› proof (unfold M_def) show "graph H' h' ∈ norm_pres_extensions E p F f" proof (rule norm_pres_extensionI2) show "linearform H' h'" using h'_def H'_def HE linearform ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› E by (rule h'_lf) show "H' ⊴ E" unfolding H'_def proof show "H ⊴ E" by fact show "vectorspace E" by fact from x'E show "lin x' ⊴ E" .. qed from H ‹F ⊴ H› HH' show FH': "F ⊴ H'" by (rule vectorspace.subspace_trans) show "graph F f ⊆ graph H' h'" proof (rule graph_extI) fix x assume x: "x ∈ F" with graphs have "f x = h x" .. also have "… = h x + 0 * xi" by simp also have "… = (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 ⋅ x' ∧ y ∈ H)" using E HE proof (rule decomp_H'_H [symmetric]) from FH x show "x ∈ H" .. from x' show "x' ≠ 0" . show "x' ∉ H" by fact show "x' ∈ E" by fact qed also have "(let (y, a) = (SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H) in h y + a * xi) = h' x" by (simp only: h'_def) finally show "f x = h' x" . next from FH' show "F ⊆ H'" .. qed show "∀x ∈ H'. h' x ≤ p x" using h'_def H'_def ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› E HE ‹seminorm E p› linearform and hp xi by (rule h'_norm_pres) qed qed ultimately show ?thesis .. qed then have "¬ (∀x ∈ M. g ⊆ x ⟶ g = x)" by simp ― ‹So the graph ‹g› of ‹h› cannot be maximal. Contradiction! ┈› with gx show "H = E" by contradiction qed from HE_eq and linearform have "linearform E h" by (simp only:) moreover have "∀x ∈ F. h x = f x" proof fix x assume "x ∈ F" with graphs have "f x = h x" .. then show "h x = f x" .. qed moreover from HE_eq and hp have "∀x ∈ E. h x ≤ 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 ‹f› and a seminorm ‹p› 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} ‹∀x ∈ H. ¦h x¦ ≤ p x› & and & ‹∀x ∈ H. h x ≤ 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: "∀x ∈ F. ¦f x¦ ≤ p x" shows "∃g. linearform E g ∧ (∀x ∈ F. g x = f x) ∧ (∀x ∈ E. ¦g x¦ ≤ 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 "∃g. linearform E g ∧ (∀x ∈ F. g x = f x) ∧ (∀x ∈ E. g x ≤ p x)" using E FE sn lf proof (rule Hahn_Banach) show "∀x ∈ F. f x ≤ 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 *: "∀x ∈ F. g x = f x" and **: "∀x ∈ E. g x ≤ p x" by blast have "∀x ∈ E. ¦g x¦ ≤ p x" using _ E sn lg ** proof (rule abs_ineq_iff [THEN iffD2]) show "E ⊴ E" .. qed with lg * show ?thesis by blast qed subsection ‹The Hahn-Banach Theorem for normed spaces› text ‹ Every continuous linear form ‹f› on a subspace ‹F› of a norm space ‹E›, can be extended to a continuous linear form ‹g› on ‹E› such that ‹∥f∥ = ∥g∥›. › theorem norm_Hahn_Banach: fixes V and norm ("∥_∥") fixes B defines "⋀V f. B V f ≡ {0} ∪ {¦f x¦ / ∥x∥ | x. x ≠ 0 ∧ x ∈ V}" fixes fn_norm ("∥_∥_" [0, 1000] 999) defines "⋀V f. ∥f∥V ≡ ⨆(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 "∃g. linearform E g ∧ continuous E g norm ∧ (∀x ∈ F. g x = f x) ∧ ∥g∥E = ∥f∥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 ≤ ∥f∥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 ‹p› on ‹E› as follows: ‹p x = ∥f∥ ⋅ ∥x∥›› define p where "p x = ∥f∥F * ∥x∥" for x txt ‹‹p› is a seminorm on ‹E›:› have q: "seminorm E p" proof fix x y a assume x: "x ∈ E" and y: "y ∈ E" txt ‹‹p› is positive definite:› have "0 ≤ ∥f∥F" by (rule ge_zero) moreover from x have "0 ≤ ∥x∥" .. ultimately show "0 ≤ p x" by (simp add: p_def zero_le_mult_iff) txt ‹‹p› is absolutely homogeneous:› show "p (a ⋅ x) = ¦a¦ * p x" proof - have "p (a ⋅ x) = ∥f∥F * ∥a ⋅ x∥" by (simp only: p_def) also from x have "∥a ⋅ x∥ = ¦a¦ * ∥x∥" by (rule abs_homogenous) also have "∥f∥F * (¦a¦ * ∥x∥) = ¦a¦ * (∥f∥F * ∥x∥)" by simp also have "… = ¦a¦ * p x" by (simp only: p_def) finally show ?thesis . qed txt ‹Furthermore, ‹p› is subadditive:› show "p (x + y) ≤ p x + p y" proof - have "p (x + y) = ∥f∥F * ∥x + y∥" by (simp only: p_def) also have a: "0 ≤ ∥f∥F" by (rule ge_zero) from x y have "∥x + y∥ ≤ ∥x∥ + ∥y∥" .. with a have " ∥f∥F * ∥x + y∥ ≤ ∥f∥F * (∥x∥ + ∥y∥)" by (simp add: mult_left_mono) also have "… = ∥f∥F * ∥x∥ + ∥f∥F * ∥y∥" by (simp only: distrib_left) also have "… = p x + p y" by (simp only: p_def) finally show ?thesis . qed qed txt ‹‹f› is bounded by ‹p›.› have "∀x ∈ F. ¦f x¦ ≤ p x" proof fix x assume "x ∈ F" with ‹continuous F f norm› and linearform show "¦f x¦ ≤ 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 ‹p› is a seminorm and ‹f› is bounded by ‹p› we can apply the Hahn-Banach Theorem for real vector spaces. So ‹f› can be extended in a norm-preserving way to some function ‹g› on the whole vector space ‹E›.› with E FE linearform q obtain g where linearformE: "linearform E g" and a: "∀x ∈ F. g x = f x" and b: "∀x ∈ E. ¦g x¦ ≤ p x" by (rule abs_Hahn_Banach [elim_format]) iprover txt ‹We furthermore have to show that ‹g› is also continuous:› have g_cont: "continuous E g norm" using linearformE proof fix x assume "x ∈ E" with b show "¦g x¦ ≤ ∥f∥F * ∥x∥" by (simp only: p_def) qed txt ‹To complete the proof, we show that ‹∥g∥ = ∥f∥›.› have "∥g∥E = ∥f∥F" proof (rule order_antisym) txt ‹ First we show ‹∥g∥ ≤ ∥f∥›. The function norm ‹∥g∥› is defined as the smallest ‹c ∈ ℝ› such that \begin{center} \begin{tabular}{l} ‹∀x ∈ E. ¦g x¦ ≤ c ⋅ ∥x∥› \end{tabular} \end{center} ⇤ Furthermore holds \begin{center} \begin{tabular}{l} ‹∀x ∈ E. ¦g x¦ ≤ ∥f∥ ⋅ ∥x∥› \end{tabular} \end{center} › from g_cont _ ge_zero show "∥g∥E ≤ ∥f∥F" proof fix x assume "x ∈ E" with b show "¦g x¦ ≤ ∥f∥F * ∥x∥" by (simp only: p_def) qed txt ‹The other direction is achieved by a similar argument.› show "∥f∥F ≤ ∥g∥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 ∈ F" show "¦f x¦ ≤ ∥g∥E * ∥x∥" proof - from a x have "g x = f x" .. then have "¦f x¦ = ¦g x¦" by (simp only:) also from g_cont have "… ≤ ∥g∥E * ∥x∥" proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def]) from FE x show "x ∈ E" .. qed finally show ?thesis . qed next show "0 ≤ ∥g∥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