author | wenzelm |
Wed, 05 Mar 2008 21:24:03 +0100 | |
changeset 26199 | 04817a8802f2 |
parent 25762 | c03e9d04b3e4 |
child 26821 | 05fd4be26c4d |
permissions | -rw-r--r-- |
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(* Title: HOL/Real/HahnBanach/Subspace.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, TU Munich |
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*) |
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header {* Subspaces *} |
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theory Subspace imports VectorSpace begin |
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subsection {* Definition *} |
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text {* |
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A non-empty subset @{text U} of a vector space @{text V} is a |
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\emph{subspace} of @{text V}, iff @{text U} is closed under addition |
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and scalar multiplication. |
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*} |
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locale subspace = var U + var V + |
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constrains U :: "'a\<Colon>{minus, plus, zero, uminus} set" |
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assumes non_empty [iff, intro]: "U \<noteq> {}" |
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and subset [iff]: "U \<subseteq> V" |
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and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U" |
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and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U" |
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notation (symbols) |
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subspace (infix "\<unlhd>" 50) |
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declare vectorspace.intro [intro?] subspace.intro [intro?] |
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lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V" |
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by (rule subspace.subset) |
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lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V" |
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using subset by blast |
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lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V" |
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by (rule subspace.subsetD) |
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lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V" |
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by (rule subspace.subsetD) |
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lemma (in subspace) diff_closed [iff]: |
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includes vectorspace |
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shows "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x - y \<in> U" |
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by (simp add: diff_eq1 negate_eq1) |
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text {* |
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\medskip Similar as for linear spaces, the existence of the zero |
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element in every subspace follows from the non-emptiness of the |
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carrier set and by vector space laws. |
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*} |
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||
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lemma (in subspace) zero [intro]: |
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includes vectorspace |
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shows "0 \<in> U" |
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proof - |
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have "U \<noteq> {}" by (rule U_V.non_empty) |
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then obtain x where x: "x \<in> U" by blast |
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hence "x \<in> V" .. hence "0 = x - x" by simp |
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also from `vectorspace V` x x have "... \<in> U" by (rule U_V.diff_closed) |
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finally show ?thesis . |
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qed |
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lemma (in subspace) neg_closed [iff]: |
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includes vectorspace |
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shows "x \<in> U \<Longrightarrow> - x \<in> U" |
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by (simp add: negate_eq1) |
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text {* \medskip Further derived laws: every subspace is a vector space. *} |
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lemma (in subspace) vectorspace [iff]: |
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includes vectorspace |
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shows "vectorspace U" |
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proof |
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show "U \<noteq> {}" .. |
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fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U" |
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fix a b :: real |
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from x y show "x + y \<in> U" by simp |
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from x show "a \<cdot> x \<in> U" by simp |
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from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac) |
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from x y show "x + y = y + x" by (simp add: add_ac) |
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from x show "x - x = 0" by simp |
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from x show "0 + x = x" by simp |
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from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib) |
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from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib) |
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from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc) |
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from x show "1 \<cdot> x = x" by simp |
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from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1) |
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from x y show "x - y = x + - y" by (simp add: diff_eq1) |
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qed |
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text {* The subspace relation is reflexive. *} |
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lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V" |
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proof |
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show "V \<noteq> {}" .. |
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show "V \<subseteq> V" .. |
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fix x y assume x: "x \<in> V" and y: "y \<in> V" |
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fix a :: real |
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from x y show "x + y \<in> V" by simp |
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from x show "a \<cdot> x \<in> V" by simp |
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qed |
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text {* The subspace relation is transitive. *} |
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lemma (in vectorspace) subspace_trans [trans]: |
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"U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W" |
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proof |
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assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W" |
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from uv show "U \<noteq> {}" by (rule subspace.non_empty) |
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show "U \<subseteq> W" |
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proof - |
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from uv have "U \<subseteq> V" by (rule subspace.subset) |
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also from vw have "V \<subseteq> W" by (rule subspace.subset) |
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finally show ?thesis . |
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qed |
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fix x y assume x: "x \<in> U" and y: "y \<in> U" |
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from uv and x y show "x + y \<in> U" by (rule subspace.add_closed) |
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from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed) |
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qed |
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subsection {* Linear closure *} |
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text {* |
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The \emph{linear closure} of a vector @{text x} is the set of all |
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scalar multiples of @{text x}. |
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*} |
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definition |
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lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set" where |
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"lin x = {a \<cdot> x | a. True}" |
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lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x" |
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by (unfold lin_def) blast |
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lemma linI' [iff]: "a \<cdot> x \<in> lin x" |
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by (unfold lin_def) blast |
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lemma linE [elim]: |
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"x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C" |
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by (unfold lin_def) blast |
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text {* Every vector is contained in its linear closure. *} |
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lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x" |
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proof - |
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assume "x \<in> V" |
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hence "x = 1 \<cdot> x" by simp |
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also have "\<dots> \<in> lin x" .. |
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finally show ?thesis . |
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qed |
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lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x" |
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proof |
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assume "x \<in> V" |
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thus "0 = 0 \<cdot> x" by simp |
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qed |
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text {* Any linear closure is a subspace. *} |
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lemma (in vectorspace) lin_subspace [intro]: |
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"x \<in> V \<Longrightarrow> lin x \<unlhd> V" |
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proof |
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assume x: "x \<in> V" |
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thus "lin x \<noteq> {}" by (auto simp add: x_lin_x) |
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show "lin x \<subseteq> V" |
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proof |
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fix x' assume "x' \<in> lin x" |
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then obtain a where "x' = a \<cdot> x" .. |
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with x show "x' \<in> V" by simp |
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qed |
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fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x" |
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show "x' + x'' \<in> lin x" |
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proof - |
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from x' obtain a' where "x' = a' \<cdot> x" .. |
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moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" .. |
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ultimately have "x' + x'' = (a' + a'') \<cdot> x" |
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using x by (simp add: distrib) |
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also have "\<dots> \<in> lin x" .. |
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finally show ?thesis . |
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qed |
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fix a :: real |
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show "a \<cdot> x' \<in> lin x" |
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proof - |
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from x' obtain a' where "x' = a' \<cdot> x" .. |
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with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc) |
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also have "\<dots> \<in> lin x" .. |
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finally show ?thesis . |
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qed |
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qed |
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text {* Any linear closure is a vector space. *} |
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lemma (in vectorspace) lin_vectorspace [intro]: |
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assumes "x \<in> V" |
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shows "vectorspace (lin x)" |
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proof - |
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from `x \<in> V` have "subspace (lin x) V" |
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by (rule lin_subspace) |
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from this and vectorspace_axioms show ?thesis |
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by (rule subspace.vectorspace) |
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qed |
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||
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subsection {* Sum of two vectorspaces *} |
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text {* |
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The \emph{sum} of two vectorspaces @{text U} and @{text V} is the |
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set of all sums of elements from @{text U} and @{text V}. |
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*} |
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instance set :: (plus) plus .. |
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defs (overloaded) |
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sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}" |
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lemma sumE [elim]: |
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"x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C" |
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by (unfold sum_def) blast |
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lemma sumI [intro]: |
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"u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V" |
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by (unfold sum_def) blast |
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lemma sumI' [intro]: |
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"u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V" |
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by (unfold sum_def) blast |
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text {* @{text U} is a subspace of @{text "U + V"}. *} |
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lemma subspace_sum1 [iff]: |
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includes vectorspace U + vectorspace V |
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shows "U \<unlhd> U + V" |
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proof |
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show "U \<noteq> {}" .. |
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show "U \<subseteq> U + V" |
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proof |
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fix x assume x: "x \<in> U" |
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moreover have "0 \<in> V" .. |
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ultimately have "x + 0 \<in> U + V" .. |
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with x show "x \<in> U + V" by simp |
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qed |
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fix x y assume x: "x \<in> U" and "y \<in> U" |
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thus "x + y \<in> U" by simp |
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from x show "\<And>a. a \<cdot> x \<in> U" by simp |
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qed |
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text {* The sum of two subspaces is again a subspace. *} |
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lemma sum_subspace [intro?]: |
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includes subspace U E + vectorspace E + subspace V E |
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shows "U + V \<unlhd> E" |
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proof |
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have "0 \<in> U + V" |
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proof |
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show "0 \<in> U" using `vectorspace E` .. |
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show "0 \<in> V" using `vectorspace E` .. |
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show "(0::'a) = 0 + 0" by simp |
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qed |
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thus "U + V \<noteq> {}" by blast |
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show "U + V \<subseteq> E" |
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proof |
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fix x assume "x \<in> U + V" |
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then obtain u v where "x = u + v" and |
272 |
"u \<in> U" and "v \<in> V" .. |
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then show "x \<in> E" by simp |
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qed |
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fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V" |
276 |
show "x + y \<in> U + V" |
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proof - |
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from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" .. |
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moreover |
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from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" .. |
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ultimately |
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have "ux + uy \<in> U" |
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and "vx + vy \<in> V" |
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and "x + y = (ux + uy) + (vx + vy)" |
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using x y by (simp_all add: add_ac) |
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thus ?thesis .. |
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qed |
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fix a show "a \<cdot> x \<in> U + V" |
289 |
proof - |
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from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" .. |
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hence "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V" |
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and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib) |
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thus ?thesis .. |
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qed |
295 |
qed |
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text{* The sum of two subspaces is a vectorspace. *} |
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lemma sum_vs [intro?]: |
300 |
"U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)" |
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by (rule subspace.vectorspace) (rule sum_subspace) |
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subsection {* Direct sums *} |
7808 | 305 |
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text {* |
307 |
The sum of @{text U} and @{text V} is called \emph{direct}, iff the |
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zero element is the only common element of @{text U} and @{text |
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V}. For every element @{text x} of the direct sum of @{text U} and |
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@{text V} the decomposition in @{text "x = u + v"} with |
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@{text "u \<in> U"} and @{text "v \<in> V"} is unique. |
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*} |
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lemma decomp: |
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includes vectorspace E + subspace U E + subspace V E |
316 |
assumes direct: "U \<inter> V = {0}" |
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and u1: "u1 \<in> U" and u2: "u2 \<in> U" |
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and v1: "v1 \<in> V" and v2: "v2 \<in> V" |
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and sum: "u1 + v1 = u2 + v2" |
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shows "u1 = u2 \<and> v1 = v2" |
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proof |
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have U: "vectorspace U" |
323 |
using `subspace U E` `vectorspace E` by (rule subspace.vectorspace) |
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have V: "vectorspace V" |
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325 |
using `subspace V E` `vectorspace E` by (rule subspace.vectorspace) |
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from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1" |
327 |
by (simp add: add_diff_swap) |
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from u1 u2 have u: "u1 - u2 \<in> U" |
|
329 |
by (rule vectorspace.diff_closed [OF U]) |
|
330 |
with eq have v': "v2 - v1 \<in> U" by (simp only:) |
|
331 |
from v2 v1 have v: "v2 - v1 \<in> V" |
|
332 |
by (rule vectorspace.diff_closed [OF V]) |
|
333 |
with eq have u': " u1 - u2 \<in> V" by (simp only:) |
|
10687 | 334 |
|
9035 | 335 |
show "u1 = u2" |
13515 | 336 |
proof (rule add_minus_eq) |
23378 | 337 |
from u1 show "u1 \<in> E" .. |
338 |
from u2 show "u2 \<in> E" .. |
|
339 |
from u u' and direct show "u1 - u2 = 0" by blast |
|
9035 | 340 |
qed |
341 |
show "v1 = v2" |
|
13515 | 342 |
proof (rule add_minus_eq [symmetric]) |
23378 | 343 |
from v1 show "v1 \<in> E" .. |
344 |
from v2 show "v2 \<in> E" .. |
|
345 |
from v v' and direct show "v2 - v1 = 0" by blast |
|
9035 | 346 |
qed |
347 |
qed |
|
7656 | 348 |
|
10687 | 349 |
text {* |
350 |
An application of the previous lemma will be used in the proof of |
|
351 |
the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any |
|
352 |
element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a |
|
353 |
vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"} |
|
354 |
the components @{text "y \<in> H"} and @{text a} are uniquely |
|
355 |
determined. |
|
356 |
*} |
|
7917 | 357 |
|
10687 | 358 |
lemma decomp_H': |
13547 | 359 |
includes vectorspace E + subspace H E |
13515 | 360 |
assumes y1: "y1 \<in> H" and y2: "y2 \<in> H" |
361 |
and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
|
362 |
and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'" |
|
363 |
shows "y1 = y2 \<and> a1 = a2" |
|
9035 | 364 |
proof |
9374 | 365 |
have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'" |
10687 | 366 |
proof (rule decomp) |
367 |
show "a1 \<cdot> x' \<in> lin x'" .. |
|
9374 | 368 |
show "a2 \<cdot> x' \<in> lin x'" .. |
13515 | 369 |
show "H \<inter> lin x' = {0}" |
9035 | 370 |
proof |
10687 | 371 |
show "H \<inter> lin x' \<subseteq> {0}" |
13515 | 372 |
proof |
373 |
fix x assume x: "x \<in> H \<inter> lin x'" |
|
374 |
then obtain a where xx': "x = a \<cdot> x'" |
|
375 |
by blast |
|
376 |
have "x = 0" |
|
377 |
proof cases |
|
378 |
assume "a = 0" |
|
379 |
with xx' and x' show ?thesis by simp |
|
380 |
next |
|
381 |
assume a: "a \<noteq> 0" |
|
382 |
from x have "x \<in> H" .. |
|
383 |
with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp |
|
384 |
with a and x' have "x' \<in> H" by (simp add: mult_assoc2) |
|
23378 | 385 |
with `x' \<notin> H` show ?thesis by contradiction |
13515 | 386 |
qed |
387 |
thus "x \<in> {0}" .. |
|
9035 | 388 |
qed |
10687 | 389 |
show "{0} \<subseteq> H \<inter> lin x'" |
9035 | 390 |
proof - |
23378 | 391 |
have "0 \<in> H" using `vectorspace E` .. |
392 |
moreover have "0 \<in> lin x'" using `x' \<in> E` .. |
|
13515 | 393 |
ultimately show ?thesis by blast |
9035 | 394 |
qed |
395 |
qed |
|
23378 | 396 |
show "lin x' \<unlhd> E" using `x' \<in> E` .. |
397 |
qed (rule `vectorspace E`, rule `subspace H E`, rule y1, rule y2, rule eq) |
|
13515 | 398 |
thus "y1 = y2" .. |
399 |
from c have "a1 \<cdot> x' = a2 \<cdot> x'" .. |
|
400 |
with x' show "a1 = a2" by (simp add: mult_right_cancel) |
|
9035 | 401 |
qed |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
402 |
|
10687 | 403 |
text {* |
404 |
Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a |
|
405 |
vectorspace @{text H} and the linear closure of @{text x'} the |
|
406 |
components @{text "y \<in> H"} and @{text a} are unique, it follows from |
|
407 |
@{text "y \<in> H"} that @{text "a = 0"}. |
|
408 |
*} |
|
7917 | 409 |
|
10687 | 410 |
lemma decomp_H'_H: |
13547 | 411 |
includes vectorspace E + subspace H E |
13515 | 412 |
assumes t: "t \<in> H" |
413 |
and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
|
414 |
shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)" |
|
415 |
proof (rule, simp_all only: split_paired_all split_conv) |
|
416 |
from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp |
|
417 |
fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H" |
|
418 |
have "y = t \<and> a = 0" |
|
419 |
proof (rule decomp_H') |
|
420 |
from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp |
|
421 |
from ya show "y \<in> H" .. |
|
23378 | 422 |
qed (rule `vectorspace E`, rule `subspace H E`, rule t, (rule x')+) |
13515 | 423 |
with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp |
424 |
qed |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
425 |
|
10687 | 426 |
text {* |
427 |
The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"} |
|
428 |
are unique, so the function @{text h'} defined by |
|
429 |
@{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite. |
|
430 |
*} |
|
7917 | 431 |
|
9374 | 432 |
lemma h'_definite: |
13515 | 433 |
includes var H |
434 |
assumes h'_def: |
|
435 |
"h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H) |
|
436 |
in (h y) + a * xi)" |
|
437 |
and x: "x = y + a \<cdot> x'" |
|
13547 | 438 |
includes vectorspace E + subspace H E |
13515 | 439 |
assumes y: "y \<in> H" |
440 |
and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" |
|
441 |
shows "h' x = h y + a * xi" |
|
10687 | 442 |
proof - |
13515 | 443 |
from x y x' have "x \<in> H + lin x'" by auto |
444 |
have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p") |
|
18689
a50587cd8414
prefer ex1I over ex_ex1I in single-step reasoning;
wenzelm
parents:
16417
diff
changeset
|
445 |
proof (rule ex_ex1I) |
13515 | 446 |
from x y show "\<exists>p. ?P p" by blast |
447 |
fix p q assume p: "?P p" and q: "?P q" |
|
448 |
show "p = q" |
|
9035 | 449 |
proof - |
13515 | 450 |
from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H" |
451 |
by (cases p) simp |
|
452 |
from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H" |
|
453 |
by (cases q) simp |
|
454 |
have "fst p = fst q \<and> snd p = snd q" |
|
455 |
proof (rule decomp_H') |
|
456 |
from xp show "fst p \<in> H" .. |
|
457 |
from xq show "fst q \<in> H" .. |
|
458 |
from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'" |
|
459 |
by simp |
|
23378 | 460 |
qed (rule `vectorspace E`, rule `subspace H E`, (rule x')+) |
13515 | 461 |
thus ?thesis by (cases p, cases q) simp |
9035 | 462 |
qed |
463 |
qed |
|
10687 | 464 |
hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)" |
13515 | 465 |
by (rule some1_equality) (simp add: x y) |
466 |
with h'_def show "h' x = h y + a * xi" by (simp add: Let_def) |
|
9035 | 467 |
qed |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
468 |
|
10687 | 469 |
end |