(* Title: HOL/Multivariate_Analysis/Operator_Norm.thy
Author: Amine Chaieb, University of Cambridge
Author: Brian Huffman
*)
section \<open>Operator Norm\<close>
theory Operator_Norm
imports Complex_Main
begin
text \<open>This formulation yields zero if @{text 'a} is the trivial vector space.\<close>
definition onorm :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> real"
where "onorm f = (SUP x. norm (f x) / norm x)"
lemma onorm_bound:
assumes "0 \<le> b" and "\<And>x. norm (f x) \<le> b * norm x"
shows "onorm f \<le> b"
unfolding onorm_def
proof (rule cSUP_least)
fix x
show "norm (f x) / norm x \<le> b"
using assms by (cases "x = 0") (simp_all add: pos_divide_le_eq)
qed simp
text \<open>In non-trivial vector spaces, the first assumption is redundant.\<close>
lemma onorm_le:
fixes f :: "'a::{real_normed_vector, perfect_space} \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>x. norm (f x) \<le> b * norm x"
shows "onorm f \<le> b"
proof (rule onorm_bound [OF _ assms])
have "{0::'a} \<noteq> UNIV" by (metis not_open_singleton open_UNIV)
then obtain a :: 'a where "a \<noteq> 0" by fast
have "0 \<le> b * norm a"
by (rule order_trans [OF norm_ge_zero assms])
with \<open>a \<noteq> 0\<close> show "0 \<le> b"
by (simp add: zero_le_mult_iff)
qed
lemma le_onorm:
assumes "bounded_linear f"
shows "norm (f x) / norm x \<le> onorm f"
proof -
interpret f: bounded_linear f by fact
obtain b where "0 \<le> b" and "\<forall>x. norm (f x) \<le> norm x * b"
using f.nonneg_bounded by auto
then have "\<forall>x. norm (f x) / norm x \<le> b"
by (clarify, case_tac "x = 0",
simp_all add: f.zero pos_divide_le_eq mult.commute)
then have "bdd_above (range (\<lambda>x. norm (f x) / norm x))"
unfolding bdd_above_def by fast
with UNIV_I show ?thesis
unfolding onorm_def by (rule cSUP_upper)
qed
lemma onorm:
assumes "bounded_linear f"
shows "norm (f x) \<le> onorm f * norm x"
proof -
interpret f: bounded_linear f by fact
show ?thesis
proof (cases)
assume "x = 0"
then show ?thesis by (simp add: f.zero)
next
assume "x \<noteq> 0"
have "norm (f x) / norm x \<le> onorm f"
by (rule le_onorm [OF assms])
then show "norm (f x) \<le> onorm f * norm x"
by (simp add: pos_divide_le_eq \<open>x \<noteq> 0\<close>)
qed
qed
lemma onorm_pos_le:
assumes f: "bounded_linear f"
shows "0 \<le> onorm f"
using le_onorm [OF f, where x=0] by simp
lemma onorm_zero: "onorm (\<lambda>x. 0) = 0"
proof (rule order_antisym)
show "onorm (\<lambda>x. 0) \<le> 0"
by (simp add: onorm_bound)
show "0 \<le> onorm (\<lambda>x. 0)"
using bounded_linear_zero by (rule onorm_pos_le)
qed
lemma onorm_eq_0:
assumes f: "bounded_linear f"
shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
using onorm [OF f] by (auto simp: fun_eq_iff [symmetric] onorm_zero)
lemma onorm_pos_lt:
assumes f: "bounded_linear f"
shows "0 < onorm f \<longleftrightarrow> \<not> (\<forall>x. f x = 0)"
by (simp add: less_le onorm_pos_le [OF f] onorm_eq_0 [OF f])
lemma onorm_compose:
assumes f: "bounded_linear f"
assumes g: "bounded_linear g"
shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
proof (rule onorm_bound)
show "0 \<le> onorm f * onorm g"
by (intro mult_nonneg_nonneg onorm_pos_le f g)
next
fix x
have "norm (f (g x)) \<le> onorm f * norm (g x)"
by (rule onorm [OF f])
also have "onorm f * norm (g x) \<le> onorm f * (onorm g * norm x)"
by (rule mult_left_mono [OF onorm [OF g] onorm_pos_le [OF f]])
finally show "norm ((f \<circ> g) x) \<le> onorm f * onorm g * norm x"
by (simp add: mult.assoc)
qed
lemma onorm_scaleR_lemma:
assumes f: "bounded_linear f"
shows "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
proof (rule onorm_bound)
show "0 \<le> \<bar>r\<bar> * onorm f"
by (intro mult_nonneg_nonneg onorm_pos_le abs_ge_zero f)
next
fix x
have "\<bar>r\<bar> * norm (f x) \<le> \<bar>r\<bar> * (onorm f * norm x)"
by (intro mult_left_mono onorm abs_ge_zero f)
then show "norm (r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f * norm x"
by (simp only: norm_scaleR mult.assoc)
qed
lemma onorm_scaleR:
assumes f: "bounded_linear f"
shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
proof (cases "r = 0")
assume "r \<noteq> 0"
show ?thesis
proof (rule order_antisym)
show "onorm (\<lambda>x. r *\<^sub>R f x) \<le> \<bar>r\<bar> * onorm f"
using f by (rule onorm_scaleR_lemma)
next
have "bounded_linear (\<lambda>x. r *\<^sub>R f x)"
using bounded_linear_scaleR_right f by (rule bounded_linear_compose)
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)"
by (rule onorm_scaleR_lemma)
with \<open>r \<noteq> 0\<close> show "\<bar>r\<bar> * onorm f \<le> onorm (\<lambda>x. r *\<^sub>R f x)"
by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
qed
qed (simp add: onorm_zero)
lemma onorm_neg:
shows "onorm (\<lambda>x. - f x) = onorm f"
unfolding onorm_def by simp
lemma onorm_triangle:
assumes f: "bounded_linear f"
assumes g: "bounded_linear g"
shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
proof (rule onorm_bound)
show "0 \<le> onorm f + onorm g"
by (intro add_nonneg_nonneg onorm_pos_le f g)
next
fix x
have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
by (rule norm_triangle_ineq)
also have "norm (f x) + norm (g x) \<le> onorm f * norm x + onorm g * norm x"
by (intro add_mono onorm f g)
finally show "norm (f x + g x) \<le> (onorm f + onorm g) * norm x"
by (simp only: distrib_right)
qed
lemma onorm_triangle_le:
assumes "bounded_linear f"
assumes "bounded_linear g"
assumes "onorm f + onorm g \<le> e"
shows "onorm (\<lambda>x. f x + g x) \<le> e"
using assms by (rule onorm_triangle [THEN order_trans])
lemma onorm_triangle_lt:
assumes "bounded_linear f"
assumes "bounded_linear g"
assumes "onorm f + onorm g < e"
shows "onorm (\<lambda>x. f x + g x) < e"
using assms by (rule onorm_triangle [THEN order_le_less_trans])
end