src/HOL/Analysis/Operator_Norm.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (4 weeks ago) changeset 69981 3dced198b9ec parent 69607 7cd977863194 child 70136 f03a01a18c6e permissions -rw-r--r--
more strict AFP properties;
```     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 text%important \<open>%whitespace\<close>
```
```    15 definition%important
```
```    16 onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real" where
```
```    17 "onorm f = (SUP x. norm (f x) / norm x)"
```
```    18
```
```    19 proposition onorm_bound:
```
```    20   assumes "0 \<le> b" and "\<And>x. norm (f x) \<le> b * norm x"
```
```    21   shows "onorm f \<le> b"
```
```    22   unfolding onorm_def
```
```    23 proof (rule cSUP_least)
```
```    24   fix x
```
```    25   show "norm (f x) / norm x \<le> b"
```
```    26     using assms by (cases "x = 0") (simp_all add: pos_divide_le_eq)
```
```    27 qed simp
```
```    28
```
```    29 text \<open>In non-trivial vector spaces, the first assumption is redundant.\<close>
```
```    30
```
```    31 lemma onorm_le:
```
```    32   fixes f :: "'a::{real_normed_vector, perfect_space} \<Rightarrow> 'b::real_normed_vector"
```
```    33   assumes "\<And>x. norm (f x) \<le> b * norm x"
```
```    34   shows "onorm f \<le> b"
```
```    35 proof (rule onorm_bound [OF _ assms])
```
```    36   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
```
```    37   then obtain a :: 'a where "a \<noteq> 0" by fast
```
```    38   have "0 \<le> b * norm a"
```
```    39     by (rule order_trans [OF norm_ge_zero assms])
```
```    40   with \<open>a \<noteq> 0\<close> show "0 \<le> b"
```
```    41     by (simp add: zero_le_mult_iff)
```
```    42 qed
```
```    43
```
```    44 lemma le_onorm:
```
```    45   assumes "bounded_linear f"
```
```    46   shows "norm (f x) / norm x \<le> onorm f"
```
```    47 proof -
```
```    48   interpret f: bounded_linear f by fact
```
```    49   obtain b where "0 \<le> b" and "\<forall>x. norm (f x) \<le> norm x * b"
```
```    50     using f.nonneg_bounded by auto
```
```    51   then have "\<forall>x. norm (f x) / norm x \<le> b"
```
```    52     by (clarify, case_tac "x = 0",
```
```    53       simp_all add: f.zero pos_divide_le_eq mult.commute)
```
```    54   then have "bdd_above (range (\<lambda>x. norm (f x) / norm x))"
```
```    55     unfolding bdd_above_def by fast
```
```    56   with UNIV_I show ?thesis
```
```    57     unfolding onorm_def by (rule cSUP_upper)
```
```    58 qed
```
```    59
```
```    60 lemma onorm:
```
```    61   assumes "bounded_linear f"
```
```    62   shows "norm (f x) \<le> onorm f * norm x"
```
```    63 proof -
```
```    64   interpret f: bounded_linear f by fact
```
```    65   show ?thesis
```
```    66   proof (cases)
```
```    67     assume "x = 0"
```
```    68     then show ?thesis by (simp add: f.zero)
```
```    69   next
```
```    70     assume "x \<noteq> 0"
```
```    71     have "norm (f x) / norm x \<le> onorm f"
```
```    72       by (rule le_onorm [OF assms])
```
```    73     then show "norm (f x) \<le> onorm f * norm x"
```
```    74       by (simp add: pos_divide_le_eq \<open>x \<noteq> 0\<close>)
```
```    75   qed
```
```    76 qed
```
```    77
```
```    78 lemma onorm_pos_le:
```
```    79   assumes f: "bounded_linear f"
```
```    80   shows "0 \<le> onorm f"
```
```    81   using le_onorm [OF f, where x=0] by simp
```
```    82
```
```    83 lemma onorm_zero: "onorm (\<lambda>x. 0) = 0"
```
```    84 proof (rule order_antisym)
```
```    85   show "onorm (\<lambda>x. 0) \<le> 0"
```
```    86     by (simp add: onorm_bound)
```
```    87   show "0 \<le> onorm (\<lambda>x. 0)"
```
```    88     using bounded_linear_zero by (rule onorm_pos_le)
```
```    89 qed
```
```    90
```
```    91 lemma onorm_eq_0:
```
```    92   assumes f: "bounded_linear f"
```
```    93   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
```
```    94   using onorm [OF f] by (auto simp: fun_eq_iff [symmetric] onorm_zero)
```
```    95
```
```    96 lemma onorm_pos_lt:
```
```    97   assumes f: "bounded_linear f"
```
```    98   shows "0 < onorm f \<longleftrightarrow> \<not> (\<forall>x. f x = 0)"
```
```    99   by (simp add: less_le onorm_pos_le [OF f] onorm_eq_0 [OF f])
```
```   100
```
```   101 lemma onorm_id_le: "onorm (\<lambda>x. x) \<le> 1"
```
```   102   by (rule onorm_bound) simp_all
```
```   103
```
```   104 lemma onorm_id: "onorm (\<lambda>x. x::'a::{real_normed_vector, perfect_space}) = 1"
```
```   105 proof (rule antisym[OF onorm_id_le])
```
```   106   have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
```
```   107   then obtain x :: 'a where "x \<noteq> 0" by fast
```
```   108   hence "1 \<le> norm x / norm x"
```
```   109     by simp
```
```   110   also have "\<dots> \<le> onorm (\<lambda>x::'a. x)"
```
```   111     by (rule le_onorm) (rule bounded_linear_ident)
```
```   112   finally show "1 \<le> onorm (\<lambda>x::'a. x)" .
```
```   113 qed
```
```   114
```
```   115 lemma onorm_compose:
```
```   116   assumes f: "bounded_linear f"
```
```   117   assumes g: "bounded_linear g"
```
```   118   shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
```
```   119 proof (rule onorm_bound)
```
```   120   show "0 \<le> onorm f * onorm g"
```
```   121     by (intro mult_nonneg_nonneg onorm_pos_le f g)
```
```   122 next
```
```   123   fix x
```
```   124   have "norm (f (g x)) \<le> onorm f * norm (g x)"
```
```   125     by (rule onorm [OF f])
```
```   126   also have "onorm f * norm (g x) \<le> onorm f * (onorm g * norm x)"
```
```   127     by (rule mult_left_mono [OF onorm [OF g] onorm_pos_le [OF f]])
```
```   128   finally show "norm ((f \<circ> g) x) \<le> onorm f * onorm g * norm x"
```
```   129     by (simp add: mult.assoc)
```
```   130 qed
```
```   131
```
```   132 lemma onorm_scaleR_lemma:
```
```   133   assumes f: "bounded_linear f"
```
```   134   shows "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
```
```   135 proof (rule onorm_bound)
```
```   136   show "0 \<le> \<bar>r\<bar> * onorm f"
```
```   137     by (intro mult_nonneg_nonneg onorm_pos_le abs_ge_zero f)
```
```   138 next
```
```   139   fix x
```
```   140   have "\<bar>r\<bar> * norm (f x) \<le> \<bar>r\<bar> * (onorm f * norm x)"
```
```   141     by (intro mult_left_mono onorm abs_ge_zero f)
```
```   142   then show "norm (r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f * norm x"
```
```   143     by (simp only: norm_scaleR mult.assoc)
```
```   144 qed
```
```   145
```
```   146 lemma onorm_scaleR:
```
```   147   assumes f: "bounded_linear f"
```
```   148   shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
```
```   149 proof (cases "r = 0")
```
```   150   assume "r \<noteq> 0"
```
```   151   show ?thesis
```
```   152   proof (rule order_antisym)
```
```   153     show "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
```
```   154       using f by (rule onorm_scaleR_lemma)
```
```   155   next
```
```   156     have "bounded_linear (\<lambda>x. r *\<^sub>R f x)"
```
```   157       using bounded_linear_scaleR_right f by (rule bounded_linear_compose)
```
```   158     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)"
```
```   159       by (rule onorm_scaleR_lemma)
```
```   160     with \<open>r \<noteq> 0\<close> show "\<bar>r\<bar> * onorm f \<le> onorm (\<lambda>x. r *\<^sub>R f x)"
```
```   161       by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
```
```   162   qed
```
```   163 qed (simp add: onorm_zero)
```
```   164
```
```   165 lemma onorm_scaleR_left_lemma:
```
```   166   assumes r: "bounded_linear r"
```
```   167   shows "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
```
```   168 proof (rule onorm_bound)
```
```   169   fix x
```
```   170   have "norm (r x *\<^sub>R f) = norm (r x) * norm f"
```
```   171     by simp
```
```   172   also have "\<dots> \<le> onorm r * norm x * norm f"
```
```   173     by (intro mult_right_mono onorm r norm_ge_zero)
```
```   174   finally show "norm (r x *\<^sub>R f) \<le> onorm r * norm f * norm x"
```
```   175     by (simp add: ac_simps)
```
```   176 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le r)
```
```   177
```
```   178 lemma onorm_scaleR_left:
```
```   179   assumes f: "bounded_linear r"
```
```   180   shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
```
```   181 proof (cases "f = 0")
```
```   182   assume "f \<noteq> 0"
```
```   183   show ?thesis
```
```   184   proof (rule order_antisym)
```
```   185     show "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
```
```   186       using f by (rule onorm_scaleR_left_lemma)
```
```   187   next
```
```   188     have bl1: "bounded_linear (\<lambda>x. r x *\<^sub>R f)"
```
```   189       by (metis bounded_linear_scaleR_const f)
```
```   190     have "bounded_linear (\<lambda>x. r x * norm f)"
```
```   191       by (metis bounded_linear_mult_const f)
```
```   192     from onorm_scaleR_left_lemma[OF this, of "inverse (norm f)"]
```
```   193     have "onorm r \<le> onorm (\<lambda>x. r x * norm f) * inverse (norm f)"
```
```   194       using \<open>f \<noteq> 0\<close>
```
```   195       by (simp add: inverse_eq_divide)
```
```   196     also have "onorm (\<lambda>x. r x * norm f) \<le> onorm (\<lambda>x. r x *\<^sub>R f)"
```
```   197       by (rule onorm_bound)
```
```   198         (auto simp: abs_mult bl1 onorm_pos_le intro!: order_trans[OF _ onorm])
```
```   199     finally show "onorm r * norm f \<le> onorm (\<lambda>x. r x *\<^sub>R f)"
```
```   200       using \<open>f \<noteq> 0\<close>
```
```   201       by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
```
```   202   qed
```
```   203 qed (simp add: onorm_zero)
```
```   204
```
```   205 lemma onorm_neg:
```
```   206   shows "onorm (\<lambda>x. - f x) = onorm f"
```
```   207   unfolding onorm_def by simp
```
```   208
```
```   209 lemma onorm_triangle:
```
```   210   assumes f: "bounded_linear f"
```
```   211   assumes g: "bounded_linear g"
```
```   212   shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
```
```   213 proof (rule onorm_bound)
```
```   214   show "0 \<le> onorm f + onorm g"
```
```   215     by (intro add_nonneg_nonneg onorm_pos_le f g)
```
```   216 next
```
```   217   fix x
```
```   218   have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   219     by (rule norm_triangle_ineq)
```
```   220   also have "norm (f x) + norm (g x) \<le> onorm f * norm x + onorm g * norm x"
```
```   221     by (intro add_mono onorm f g)
```
```   222   finally show "norm (f x + g x) \<le> (onorm f + onorm g) * norm x"
```
```   223     by (simp only: distrib_right)
```
```   224 qed
```
```   225
```
```   226 lemma onorm_triangle_le:
```
```   227   assumes "bounded_linear f"
```
```   228   assumes "bounded_linear g"
```
```   229   assumes "onorm f + onorm g \<le> e"
```
```   230   shows "onorm (\<lambda>x. f x + g x) \<le> e"
```
```   231   using assms by (rule onorm_triangle [THEN order_trans])
```
```   232
```
```   233 lemma onorm_triangle_lt:
```
```   234   assumes "bounded_linear f"
```
```   235   assumes "bounded_linear g"
```
```   236   assumes "onorm f + onorm g < e"
```
```   237   shows "onorm (\<lambda>x. f x + g x) < e"
```
```   238   using assms by (rule onorm_triangle [THEN order_le_less_trans])
```
```   239
```
```   240 lemma onorm_sum:
```
```   241   assumes "finite S"
```
```   242   assumes "\<And>s. s \<in> S \<Longrightarrow> bounded_linear (f s)"
```
```   243   shows "onorm (\<lambda>x. sum (\<lambda>s. f s x) S) \<le> sum (\<lambda>s. onorm (f s)) S"
```
```   244   using assms
```
```   245   by (induction) (auto simp: onorm_zero intro!: onorm_triangle_le bounded_linear_sum)
```
```   246
```
```   247 lemmas onorm_sum_le = onorm_sum[THEN order_trans]
```
```   248
```
```   249 end
```