src/HOL/Analysis/Operator_Norm.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63627 6ddb43c6b711 child 67685 bdff8bf0a75b permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Analysis/Operator_Norm.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3     Author:     Brian Huffman
```
```     4 *)
```
```     5
```
```     6 section \<open>Operator Norm\<close>
```
```     7
```
```     8 theory Operator_Norm
```
```     9 imports Complex_Main
```
```    10 begin
```
```    11
```
```    12 text \<open>This formulation yields zero if \<open>'a\<close> is the trivial vector space.\<close>
```
```    13
```
```    14 definition onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real"
```
```    15   where "onorm f = (SUP x. norm (f x) / norm x)"
```
```    16
```
```    17 lemma onorm_bound:
```
```    18   assumes "0 \<le> b" and "\<And>x. norm (f x) \<le> b * norm x"
```
```    19   shows "onorm f \<le> b"
```
```    20   unfolding onorm_def
```
```    21 proof (rule cSUP_least)
```
```    22   fix x
```
```    23   show "norm (f x) / norm x \<le> b"
```
```    24     using assms by (cases "x = 0") (simp_all add: pos_divide_le_eq)
```
```    25 qed simp
```
```    26
```
```    27 text \<open>In non-trivial vector spaces, the first assumption is redundant.\<close>
```
```    28
```
```    29 lemma onorm_le:
```
```    30   fixes f :: "'a::{real_normed_vector, perfect_space} \<Rightarrow> 'b::real_normed_vector"
```
```    31   assumes "\<And>x. norm (f x) \<le> b * norm x"
```
```    32   shows "onorm f \<le> b"
```
```    33 proof (rule onorm_bound [OF _ assms])
```
```    34   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
```
```    35   then obtain a :: 'a where "a \<noteq> 0" by fast
```
```    36   have "0 \<le> b * norm a"
```
```    37     by (rule order_trans [OF norm_ge_zero assms])
```
```    38   with \<open>a \<noteq> 0\<close> show "0 \<le> b"
```
```    39     by (simp add: zero_le_mult_iff)
```
```    40 qed
```
```    41
```
```    42 lemma le_onorm:
```
```    43   assumes "bounded_linear f"
```
```    44   shows "norm (f x) / norm x \<le> onorm f"
```
```    45 proof -
```
```    46   interpret f: bounded_linear f by fact
```
```    47   obtain b where "0 \<le> b" and "\<forall>x. norm (f x) \<le> norm x * b"
```
```    48     using f.nonneg_bounded by auto
```
```    49   then have "\<forall>x. norm (f x) / norm x \<le> b"
```
```    50     by (clarify, case_tac "x = 0",
```
```    51       simp_all add: f.zero pos_divide_le_eq mult.commute)
```
```    52   then have "bdd_above (range (\<lambda>x. norm (f x) / norm x))"
```
```    53     unfolding bdd_above_def by fast
```
```    54   with UNIV_I show ?thesis
```
```    55     unfolding onorm_def by (rule cSUP_upper)
```
```    56 qed
```
```    57
```
```    58 lemma onorm:
```
```    59   assumes "bounded_linear f"
```
```    60   shows "norm (f x) \<le> onorm f * norm x"
```
```    61 proof -
```
```    62   interpret f: bounded_linear f by fact
```
```    63   show ?thesis
```
```    64   proof (cases)
```
```    65     assume "x = 0"
```
```    66     then show ?thesis by (simp add: f.zero)
```
```    67   next
```
```    68     assume "x \<noteq> 0"
```
```    69     have "norm (f x) / norm x \<le> onorm f"
```
```    70       by (rule le_onorm [OF assms])
```
```    71     then show "norm (f x) \<le> onorm f * norm x"
```
```    72       by (simp add: pos_divide_le_eq \<open>x \<noteq> 0\<close>)
```
```    73   qed
```
```    74 qed
```
```    75
```
```    76 lemma onorm_pos_le:
```
```    77   assumes f: "bounded_linear f"
```
```    78   shows "0 \<le> onorm f"
```
```    79   using le_onorm [OF f, where x=0] by simp
```
```    80
```
```    81 lemma onorm_zero: "onorm (\<lambda>x. 0) = 0"
```
```    82 proof (rule order_antisym)
```
```    83   show "onorm (\<lambda>x. 0) \<le> 0"
```
```    84     by (simp add: onorm_bound)
```
```    85   show "0 \<le> onorm (\<lambda>x. 0)"
```
```    86     using bounded_linear_zero by (rule onorm_pos_le)
```
```    87 qed
```
```    88
```
```    89 lemma onorm_eq_0:
```
```    90   assumes f: "bounded_linear f"
```
```    91   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
```
```    92   using onorm [OF f] by (auto simp: fun_eq_iff [symmetric] onorm_zero)
```
```    93
```
```    94 lemma onorm_pos_lt:
```
```    95   assumes f: "bounded_linear f"
```
```    96   shows "0 < onorm f \<longleftrightarrow> \<not> (\<forall>x. f x = 0)"
```
```    97   by (simp add: less_le onorm_pos_le [OF f] onorm_eq_0 [OF f])
```
```    98
```
```    99 lemma onorm_id_le: "onorm (\<lambda>x. x) \<le> 1"
```
```   100   by (rule onorm_bound) simp_all
```
```   101
```
```   102 lemma onorm_id: "onorm (\<lambda>x. x::'a::{real_normed_vector, perfect_space}) = 1"
```
```   103 proof (rule antisym[OF onorm_id_le])
```
```   104   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
```
```   105   then obtain x :: 'a where "x \<noteq> 0" by fast
```
```   106   hence "1 \<le> norm x / norm x"
```
```   107     by simp
```
```   108   also have "\<dots> \<le> onorm (\<lambda>x::'a. x)"
```
```   109     by (rule le_onorm) (rule bounded_linear_ident)
```
```   110   finally show "1 \<le> onorm (\<lambda>x::'a. x)" .
```
```   111 qed
```
```   112
```
```   113 lemma onorm_compose:
```
```   114   assumes f: "bounded_linear f"
```
```   115   assumes g: "bounded_linear g"
```
```   116   shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
```
```   117 proof (rule onorm_bound)
```
```   118   show "0 \<le> onorm f * onorm g"
```
```   119     by (intro mult_nonneg_nonneg onorm_pos_le f g)
```
```   120 next
```
```   121   fix x
```
```   122   have "norm (f (g x)) \<le> onorm f * norm (g x)"
```
```   123     by (rule onorm [OF f])
```
```   124   also have "onorm f * norm (g x) \<le> onorm f * (onorm g * norm x)"
```
```   125     by (rule mult_left_mono [OF onorm [OF g] onorm_pos_le [OF f]])
```
```   126   finally show "norm ((f \<circ> g) x) \<le> onorm f * onorm g * norm x"
```
```   127     by (simp add: mult.assoc)
```
```   128 qed
```
```   129
```
```   130 lemma onorm_scaleR_lemma:
```
```   131   assumes f: "bounded_linear f"
```
```   132   shows "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
```
```   133 proof (rule onorm_bound)
```
```   134   show "0 \<le> \<bar>r\<bar> * onorm f"
```
```   135     by (intro mult_nonneg_nonneg onorm_pos_le abs_ge_zero f)
```
```   136 next
```
```   137   fix x
```
```   138   have "\<bar>r\<bar> * norm (f x) \<le> \<bar>r\<bar> * (onorm f * norm x)"
```
```   139     by (intro mult_left_mono onorm abs_ge_zero f)
```
```   140   then show "norm (r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f * norm x"
```
```   141     by (simp only: norm_scaleR mult.assoc)
```
```   142 qed
```
```   143
```
```   144 lemma onorm_scaleR:
```
```   145   assumes f: "bounded_linear f"
```
```   146   shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
```
```   147 proof (cases "r = 0")
```
```   148   assume "r \<noteq> 0"
```
```   149   show ?thesis
```
```   150   proof (rule order_antisym)
```
```   151     show "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
```
```   152       using f by (rule onorm_scaleR_lemma)
```
```   153   next
```
```   154     have "bounded_linear (\<lambda>x. r *\<^sub>R f x)"
```
```   155       using bounded_linear_scaleR_right f by (rule bounded_linear_compose)
```
```   156     then have "onorm (\<lambda>x. inverse r *\<^sub>R r *\<^sub>R f x) \<le> \<bar>inverse r\<bar> * onorm (\<lambda>x. r *\<^sub>R f x)"
```
```   157       by (rule onorm_scaleR_lemma)
```
```   158     with \<open>r \<noteq> 0\<close> show "\<bar>r\<bar> * onorm f \<le> onorm (\<lambda>x. r *\<^sub>R f x)"
```
```   159       by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
```
```   160   qed
```
```   161 qed (simp add: onorm_zero)
```
```   162
```
```   163 lemma onorm_scaleR_left_lemma:
```
```   164   assumes r: "bounded_linear r"
```
```   165   shows "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
```
```   166 proof (rule onorm_bound)
```
```   167   fix x
```
```   168   have "norm (r x *\<^sub>R f) = norm (r x) * norm f"
```
```   169     by simp
```
```   170   also have "\<dots> \<le> onorm r * norm x * norm f"
```
```   171     by (intro mult_right_mono onorm r norm_ge_zero)
```
```   172   finally show "norm (r x *\<^sub>R f) \<le> onorm r * norm f * norm x"
```
```   173     by (simp add: ac_simps)
```
```   174 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le r)
```
```   175
```
```   176 lemma onorm_scaleR_left:
```
```   177   assumes f: "bounded_linear r"
```
```   178   shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
```
```   179 proof (cases "f = 0")
```
```   180   assume "f \<noteq> 0"
```
```   181   show ?thesis
```
```   182   proof (rule order_antisym)
```
```   183     show "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
```
```   184       using f by (rule onorm_scaleR_left_lemma)
```
```   185   next
```
```   186     have bl1: "bounded_linear (\<lambda>x. r x *\<^sub>R f)"
```
```   187       by (metis bounded_linear_scaleR_const f)
```
```   188     have "bounded_linear (\<lambda>x. r x * norm f)"
```
```   189       by (metis bounded_linear_mult_const f)
```
```   190     from onorm_scaleR_left_lemma[OF this, of "inverse (norm f)"]
```
```   191     have "onorm r \<le> onorm (\<lambda>x. r x * norm f) * inverse (norm f)"
```
```   192       using \<open>f \<noteq> 0\<close>
```
```   193       by (simp add: inverse_eq_divide)
```
```   194     also have "onorm (\<lambda>x. r x * norm f) \<le> onorm (\<lambda>x. r x *\<^sub>R f)"
```
```   195       by (rule onorm_bound)
```
```   196         (auto simp: abs_mult bl1 onorm_pos_le intro!: order_trans[OF _ onorm])
```
```   197     finally show "onorm r * norm f \<le> onorm (\<lambda>x. r x *\<^sub>R f)"
```
```   198       using \<open>f \<noteq> 0\<close>
```
```   199       by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
```
```   200   qed
```
```   201 qed (simp add: onorm_zero)
```
```   202
```
```   203 lemma onorm_neg:
```
```   204   shows "onorm (\<lambda>x. - f x) = onorm f"
```
```   205   unfolding onorm_def by simp
```
```   206
```
```   207 lemma onorm_triangle:
```
```   208   assumes f: "bounded_linear f"
```
```   209   assumes g: "bounded_linear g"
```
```   210   shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
```
```   211 proof (rule onorm_bound)
```
```   212   show "0 \<le> onorm f + onorm g"
```
```   213     by (intro add_nonneg_nonneg onorm_pos_le f g)
```
```   214 next
```
```   215   fix x
```
```   216   have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   217     by (rule norm_triangle_ineq)
```
```   218   also have "norm (f x) + norm (g x) \<le> onorm f * norm x + onorm g * norm x"
```
```   219     by (intro add_mono onorm f g)
```
```   220   finally show "norm (f x + g x) \<le> (onorm f + onorm g) * norm x"
```
```   221     by (simp only: distrib_right)
```
```   222 qed
```
```   223
```
```   224 lemma onorm_triangle_le:
```
```   225   assumes "bounded_linear f"
```
```   226   assumes "bounded_linear g"
```
```   227   assumes "onorm f + onorm g \<le> e"
```
```   228   shows "onorm (\<lambda>x. f x + g x) \<le> e"
```
```   229   using assms by (rule onorm_triangle [THEN order_trans])
```
```   230
```
```   231 lemma onorm_triangle_lt:
```
```   232   assumes "bounded_linear f"
```
```   233   assumes "bounded_linear g"
```
```   234   assumes "onorm f + onorm g < e"
```
```   235   shows "onorm (\<lambda>x. f x + g x) < e"
```
```   236   using assms by (rule onorm_triangle [THEN order_le_less_trans])
```
```   237
```
```   238 end
```