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theory HahnBanachExtLemmas(* Title: HOL/Real/HahnBanach/HahnBanachExtLemmas.thy ID: $Id: HahnBanachExtLemmas.thy,v 1.28 2007/06/13 22:22:45 wenzelm Exp $ Author: Gertrud Bauer, TU Munich *) header {* Extending non-maximal functions *} theory HahnBanachExtLemmas imports FunctionNorm begin text {* In this section the following context is presumed. Let @{text E} be a real vector space with a seminorm @{text q} on @{text E}. @{text F} is a subspace of @{text E} and @{text f} a linear function on @{text F}. We consider a subspace @{text H} of @{text E} that is a superspace of @{text F} and a linear form @{text h} on @{text H}. @{text H} is a not equal to @{text E} and @{text "x0"} is an element in @{text "E - H"}. @{text H} is extended to the direct sum @{text "H' = H + lin x0"}, so for any @{text "x ∈ H'"} the decomposition of @{text "x = y + a · x"} with @{text "y ∈ H"} is unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y + a · ξ"} for a certain @{text ξ}. Subsequently we show some properties of this extension @{text h'} of @{text h}. \medskip This lemma will be used to show the existence of a linear extension of @{text f} (see page \pageref{ex-xi-use}). It is a consequence of the completeness of @{text \<real>}. To show \begin{center} \begin{tabular}{l} @{text "∃ξ. ∀y ∈ F. a y ≤ ξ ∧ ξ ≤ b y"} \end{tabular} \end{center} \noindent it suffices to show that \begin{center} \begin{tabular}{l} @{text "∀u ∈ F. ∀v ∈ F. a u ≤ b v"} \end{tabular} \end{center} *} lemma ex_xi: includes vectorspace F assumes r: "!!u v. u ∈ F ==> v ∈ F ==> a u ≤ b v" shows "∃xi::real. ∀y ∈ F. a y ≤ xi ∧ xi ≤ b y" proof - txt {* From the completeness of the reals follows: The set @{text "S = {a u. u ∈ F}"} has a supremum, if it is non-empty and has an upper bound. *} let ?S = "{a u | u. u ∈ F}" have "∃xi. lub ?S xi" proof (rule real_complete) have "a 0 ∈ ?S" by blast then show "∃X. X ∈ ?S" .. have "∀y ∈ ?S. y ≤ b 0" proof fix y assume y: "y ∈ ?S" then obtain u where u: "u ∈ F" and y: "y = a u" by blast from u and zero have "a u ≤ b 0" by (rule r) with y show "y ≤ b 0" by (simp only:) qed then show "∃u. ∀y ∈ ?S. y ≤ u" .. qed then obtain xi where xi: "lub ?S xi" .. { fix y assume "y ∈ F" then have "a y ∈ ?S" by blast with xi have "a y ≤ xi" by (rule lub.upper) } moreover { fix y assume y: "y ∈ F" from xi have "xi ≤ b y" proof (rule lub.least) fix au assume "au ∈ ?S" then obtain u where u: "u ∈ F" and au: "au = a u" by blast from u y have "a u ≤ b y" by (rule r) with au show "au ≤ b y" by (simp only:) qed } ultimately show "∃xi. ∀y ∈ F. a y ≤ xi ∧ xi ≤ b y" by blast qed text {* \medskip The function @{text h'} is defined as a @{text "h' x = h y + a · ξ"} where @{text "x = y + a · ξ"} is a linear extension of @{text h} to @{text H'}. *} lemma h'_lf: includes var H + var h + var E assumes h'_def: "h' ≡ λx. let (y, a) = SOME (y, a). x = y + a · x0 ∧ y ∈ H in h y + a * xi" and H'_def: "H' ≡ H + lin x0" and HE: "H \<unlhd> E" includes linearform H h assumes x0: "x0 ∉ H" "x0 ∈ E" "x0 ≠ 0" includes vectorspace E shows "linearform H' h'" proof note E = `vectorspace E` have H': "vectorspace H'" proof (unfold H'_def) from `x0 ∈ E` have "lin x0 \<unlhd> E" .. with HE show "vectorspace (H + lin x0)" using E .. qed { fix x1 x2 assume x1: "x1 ∈ H'" and x2: "x2 ∈ H'" show "h' (x1 + x2) = h' x1 + h' x2" proof - from H' x1 x2 have "x1 + x2 ∈ H'" by (rule vectorspace.add_closed) with x1 x2 obtain y y1 y2 a a1 a2 where x1x2: "x1 + x2 = y + a · x0" and y: "y ∈ H" and x1_rep: "x1 = y1 + a1 · x0" and y1: "y1 ∈ H" and x2_rep: "x2 = y2 + a2 · x0" and y2: "y2 ∈ H" by (unfold H'_def sum_def lin_def) blast have ya: "y1 + y2 = y ∧ a1 + a2 = a" using E HE _ y x0 proof (rule decomp_H') txt_raw {* \label{decomp-H-use} *} from HE y1 y2 show "y1 + y2 ∈ H" by (rule subspace.add_closed) from x0 and HE y y1 y2 have "x0 ∈ E" "y ∈ E" "y1 ∈ E" "y2 ∈ E" by auto with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) · x0 = x1 + x2" by (simp add: add_ac add_mult_distrib2) also note x1x2 finally show "(y1 + y2) + (a1 + a2) · x0 = y + a · x0" . qed from h'_def x1x2 E HE y x0 have "h' (x1 + x2) = h y + a * xi" by (rule h'_definite) also have "… = h (y1 + y2) + (a1 + a2) * xi" by (simp only: ya) also from y1 y2 have "h (y1 + y2) = h y1 + h y2" by simp also have "… + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)" by (simp add: left_distrib) also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1" by (rule h'_definite [symmetric]) also from h'_def x2_rep E HE y2 x0 have "h y2 + a2 * xi = h' x2" by (rule h'_definite [symmetric]) finally show ?thesis . qed next fix x1 c assume x1: "x1 ∈ H'" show "h' (c · x1) = c * (h' x1)" proof - from H' x1 have ax1: "c · x1 ∈ H'" by (rule vectorspace.mult_closed) with x1 obtain y a y1 a1 where cx1_rep: "c · x1 = y + a · x0" and y: "y ∈ H" and x1_rep: "x1 = y1 + a1 · x0" and y1: "y1 ∈ H" by (unfold H'_def sum_def lin_def) blast have ya: "c · y1 = y ∧ c * a1 = a" using E HE _ y x0 proof (rule decomp_H') from HE y1 show "c · y1 ∈ H" by (rule subspace.mult_closed) from x0 and HE y y1 have "x0 ∈ E" "y ∈ E" "y1 ∈ E" by auto with x1_rep have "c · y1 + (c * a1) · x0 = c · x1" by (simp add: mult_assoc add_mult_distrib1) also note cx1_rep finally show "c · y1 + (c * a1) · x0 = y + a · x0" . qed from h'_def cx1_rep E HE y x0 have "h' (c · x1) = h y + a * xi" by (rule h'_definite) also have "… = h (c · y1) + (c * a1) * xi" by (simp only: ya) also from y1 have "h (c · y1) = c * h y1" by simp also have "… + (c * a1) * xi = c * (h y1 + a1 * xi)" by (simp only: right_distrib) also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1" by (rule h'_definite [symmetric]) finally show ?thesis . qed } qed text {* \medskip The linear extension @{text h'} of @{text h} is bounded by the seminorm @{text p}. *} lemma h'_norm_pres: includes var H + var h + var E assumes h'_def: "h' ≡ λx. let (y, a) = SOME (y, a). x = y + a · x0 ∧ y ∈ H in h y + a * xi" and H'_def: "H' ≡ H + lin x0" and x0: "x0 ∉ H" "x0 ∈ E" "x0 ≠ 0" includes vectorspace E + subspace H E + seminorm E p + linearform H h assumes a: "∀y ∈ H. h y ≤ p y" and a': "∀y ∈ H. - p (y + x0) - h y ≤ xi ∧ xi ≤ p (y + x0) - h y" shows "∀x ∈ H'. h' x ≤ p x" proof note E = `vectorspace E` note HE = `subspace H E` fix x assume x': "x ∈ H'" show "h' x ≤ p x" proof - from a' have a1: "∀ya ∈ H. - p (ya + x0) - h ya ≤ xi" and a2: "∀ya ∈ H. xi ≤ p (ya + x0) - h ya" by auto from x' obtain y a where x_rep: "x = y + a · x0" and y: "y ∈ H" by (unfold H'_def sum_def lin_def) blast from y have y': "y ∈ E" .. from y have ay: "inverse a · y ∈ H" by simp from h'_def x_rep E HE y x0 have "h' x = h y + a * xi" by (rule h'_definite) also have "… ≤ p (y + a · x0)" proof (rule linorder_cases) assume z: "a = 0" then have "h y + a * xi = h y" by simp also from a y have "… ≤ p y" .. also from x0 y' z have "p y = p (y + a · x0)" by simp finally show ?thesis . next txt {* In the case @{text "a < 0"}, we use @{text "a1"} with @{text ya} taken as @{text "y / a"}: *} assume lz: "a < 0" hence nz: "a ≠ 0" by simp from a1 ay have "- p (inverse a · y + x0) - h (inverse a · y) ≤ xi" .. with lz have "a * xi ≤ a * (- p (inverse a · y + x0) - h (inverse a · y))" by (simp add: mult_left_mono_neg order_less_imp_le) also have "… = - a * (p (inverse a · y + x0)) - a * (h (inverse a · y))" by (simp add: right_diff_distrib) also from lz x0 y' have "- a * (p (inverse a · y + x0)) = p (a · (inverse a · y + x0))" by (simp add: abs_homogenous) also from nz x0 y' have "… = p (y + a · x0)" by (simp add: add_mult_distrib1 mult_assoc [symmetric]) also from nz y have "a * (h (inverse a · y)) = h y" by simp finally have "a * xi ≤ p (y + a · x0) - h y" . then show ?thesis by simp next txt {* In the case @{text "a > 0"}, we use @{text "a2"} with @{text ya} taken as @{text "y / a"}: *} assume gz: "0 < a" hence nz: "a ≠ 0" by simp from a2 ay have "xi ≤ p (inverse a · y + x0) - h (inverse a · y)" .. with gz have "a * xi ≤ a * (p (inverse a · y + x0) - h (inverse a · y))" by simp also have "... = a * p (inverse a · y + x0) - a * h (inverse a · y)" by (simp add: right_diff_distrib) also from gz x0 y' have "a * p (inverse a · y + x0) = p (a · (inverse a · y + x0))" by (simp add: abs_homogenous) also from nz x0 y' have "… = p (y + a · x0)" by (simp add: add_mult_distrib1 mult_assoc [symmetric]) also from nz y have "a * h (inverse a · y) = h y" by simp finally have "a * xi ≤ p (y + a · x0) - h y" . then show ?thesis by simp qed also from x_rep have "… = p x" by (simp only:) finally show ?thesis . qed qed end
lemma ex_xi:
[| vectorspace F; !!u v. [| u ∈ F; v ∈ F |] ==> a u ≤ b v |]
==> ∃xi. ∀y∈F. a y ≤ xi ∧ xi ≤ b y
lemma h'_lf:
[| h' == λx. let (y, a) = SOME (y, a). x = y + a · x0.0 ∧ y ∈ H in h y + a * xi;
H' == H + lin x0.0; subspace H E; linearform H h; x0.0 ∉ H; x0.0 ∈ E;
x0.0 ≠ (0::'a); vectorspace E |]
==> linearform H' h'
lemma h'_norm_pres:
[| h' == λx. let (y, a) = SOME (y, a). x = y + a · x0.0 ∧ y ∈ H in h y + a * xi;
H' == H + lin x0.0; x0.0 ∉ H; x0.0 ∈ E; x0.0 ≠ (0::'a); vectorspace E;
subspace H E; seminorm E p; linearform H h; ∀y∈H. h y ≤ p y;
∀y∈H. - p (y + x0.0) - h y ≤ xi ∧ xi ≤ p (y + x0.0) - h y |]
==> ∀x∈H'. h' x ≤ p x