src/HOL/Multivariate_Analysis/Operator_Norm.thy
author huffman
Wed Apr 28 15:05:45 2010 -0700 (2010-04-28)
changeset 36581 bbea7f52e8e1
child 36593 fb69c8cd27bd
permissions -rw-r--r--
move operator norm stuff to new theory file
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(*  Title:      Library/Operator_Norm.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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header {* Operator Norm *}
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theory Operator_Norm
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imports Euclidean_Space
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begin
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definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
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lemma norm_bound_generalize:
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  fixes f:: "real ^'n \<Rightarrow> real^'m"
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  assumes lf: "linear f"
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  shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
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proof-
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  {assume H: ?rhs
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    {fix x :: "real^'n" assume x: "norm x = 1"
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      from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
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    then have ?lhs by blast }
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  moreover
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  {assume H: ?lhs
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    from H[rule_format, of "basis arbitrary"]
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    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
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      by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
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    {fix x :: "real ^'n"
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      {assume "x = 0"
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        then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
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      moreover
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      {assume x0: "x \<noteq> 0"
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        hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
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        let ?c = "1/ norm x"
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        have "norm (?c*s x) = 1" using x0 by (simp add: n0)
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        with H have "norm (f(?c*s x)) \<le> b" by blast
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        hence "?c * norm (f x) \<le> b"
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          by (simp add: linear_cmul[OF lf])
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        hence "norm (f x) \<le> b * norm x"
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          using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
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      ultimately have "norm (f x) \<le> b * norm x" by blast}
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    then have ?rhs by blast}
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  ultimately show ?thesis by blast
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qed
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lemma onorm:
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  fixes f:: "real ^'n \<Rightarrow> real ^'m"
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  assumes lf: "linear f"
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  shows "norm (f x) <= onorm f * norm x"
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  and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
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proof-
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  {
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    let ?S = "{norm (f x) |x. norm x = 1}"
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    have Se: "?S \<noteq> {}" using  norm_basis by auto
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    from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
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      unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
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    {from Sup[OF Se b, unfolded onorm_def[symmetric]]
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      show "norm (f x) <= onorm f * norm x"
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        apply -
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        apply (rule spec[where x = x])
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        unfolding norm_bound_generalize[OF lf, symmetric]
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        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
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    {
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      show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
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        using Sup[OF Se b, unfolded onorm_def[symmetric]]
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        unfolding norm_bound_generalize[OF lf, symmetric]
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        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
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  }
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qed
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lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
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  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
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lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
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  shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
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  using onorm[OF lf]
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  apply (auto simp add: onorm_pos_le)
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  apply atomize
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  apply (erule allE[where x="0::real"])
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  using onorm_pos_le[OF lf]
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  apply arith
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  done
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lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^'m)) = norm y"
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proof-
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  let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
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  have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
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    by(auto intro: vector_choose_size set_ext)
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  show ?thesis
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    unfolding onorm_def th
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    apply (rule Sup_unique) by (simp_all  add: setle_def)
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qed
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lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
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  shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
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  unfolding onorm_eq_0[OF lf, symmetric]
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  using onorm_pos_le[OF lf] by arith
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lemma onorm_compose:
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  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
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  and lg: "linear (g::real^'k \<Rightarrow> real^'n)"
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  shows "onorm (f o g) <= onorm f * onorm g"
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  apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
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  unfolding o_def
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  apply (subst mult_assoc)
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  apply (rule order_trans)
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  apply (rule onorm(1)[OF lf])
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  apply (rule mult_mono1)
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  apply (rule onorm(1)[OF lg])
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  apply (rule onorm_pos_le[OF lf])
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  done
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lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
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  shows "onorm (\<lambda>x. - f x) \<le> onorm f"
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  using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
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  unfolding norm_minus_cancel by metis
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lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
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  shows "onorm (\<lambda>x. - f x) = onorm f"
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  using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
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  by simp
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lemma onorm_triangle:
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  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
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  shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
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  apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
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  apply (rule order_trans)
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  apply (rule norm_triangle_ineq)
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  apply (simp add: distrib)
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  apply (rule add_mono)
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  apply (rule onorm(1)[OF lf])
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  apply (rule onorm(1)[OF lg])
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  done
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lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
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  \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
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  apply (rule order_trans)
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  apply (rule onorm_triangle)
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  apply assumption+
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  done
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lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
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  ==> onorm(\<lambda>x. f x + g x) < e"
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  apply (rule order_le_less_trans)
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  apply (rule onorm_triangle)
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  by assumption+
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end