# Theory Operator_Norm

theory Operator_Norm
imports Complex_Main
```(*  Title:      HOL/Multivariate_Analysis/Operator_Norm.thy
Author:     Amine Chaieb, University of Cambridge
Author:     Brian Huffman
*)

section ‹Operator Norm›

theory Operator_Norm
imports Complex_Main
begin

text ‹This formulation yields zero if ‹'a› is the trivial vector space.›

definition onorm :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ real"
where "onorm f = (SUP x. norm (f x) / norm x)"

lemma onorm_bound:
assumes "0 ≤ b" and "⋀x. norm (f x) ≤ b * norm x"
shows "onorm f ≤ b"
unfolding onorm_def
proof (rule cSUP_least)
fix x
show "norm (f x) / norm x ≤ b"
using assms by (cases "x = 0") (simp_all add: pos_divide_le_eq)
qed simp

text ‹In non-trivial vector spaces, the first assumption is redundant.›

lemma onorm_le:
fixes f :: "'a::{real_normed_vector, perfect_space} ⇒ 'b::real_normed_vector"
assumes "⋀x. norm (f x) ≤ b * norm x"
shows "onorm f ≤ b"
proof (rule onorm_bound [OF _ assms])
have "{0::'a} ≠ UNIV" by (metis not_open_singleton open_UNIV)
then obtain a :: 'a where "a ≠ 0" by fast
have "0 ≤ b * norm a"
by (rule order_trans [OF norm_ge_zero assms])
with ‹a ≠ 0› show "0 ≤ b"
qed

lemma le_onorm:
assumes "bounded_linear f"
shows "norm (f x) / norm x ≤ onorm f"
proof -
interpret f: bounded_linear f by fact
obtain b where "0 ≤ b" and "∀x. norm (f x) ≤ norm x * b"
using f.nonneg_bounded by auto
then have "∀x. norm (f x) / norm x ≤ b"
by (clarify, case_tac "x = 0",
then have "bdd_above (range (λx. 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) ≤ 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 ≠ 0"
have "norm (f x) / norm x ≤ onorm f"
by (rule le_onorm [OF assms])
then show "norm (f x) ≤ onorm f * norm x"
by (simp add: pos_divide_le_eq ‹x ≠ 0›)
qed
qed

lemma onorm_pos_le:
assumes f: "bounded_linear f"
shows "0 ≤ onorm f"
using le_onorm [OF f, where x=0] by simp

lemma onorm_zero: "onorm (λx. 0) = 0"
proof (rule order_antisym)
show "onorm (λx. 0) ≤ 0"
show "0 ≤ onorm (λx. 0)"
using bounded_linear_zero by (rule onorm_pos_le)
qed

lemma onorm_eq_0:
assumes f: "bounded_linear f"
shows "onorm f = 0 ⟷ (∀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 ⟷ ¬ (∀x. f x = 0)"
by (simp add: less_le onorm_pos_le [OF f] onorm_eq_0 [OF f])

lemma onorm_id_le: "onorm (λx. x) ≤ 1"
by (rule onorm_bound) simp_all

lemma onorm_id: "onorm (λx. x::'a::{real_normed_vector, perfect_space}) = 1"
proof (rule antisym[OF onorm_id_le])
have "{0::'a} ≠ UNIV" by (metis not_open_singleton open_UNIV)
then obtain x :: 'a where "x ≠ 0" by fast
hence "1 ≤ norm x / norm x"
by simp
also have "… ≤ onorm (λx::'a. x)"
by (rule le_onorm) (rule bounded_linear_ident)
finally show "1 ≤ onorm (λx::'a. x)" .
qed

lemma onorm_compose:
assumes f: "bounded_linear f"
assumes g: "bounded_linear g"
shows "onorm (f ∘ g) ≤ onorm f * onorm g"
proof (rule onorm_bound)
show "0 ≤ onorm f * onorm g"
by (intro mult_nonneg_nonneg onorm_pos_le f g)
next
fix x
have "norm (f (g x)) ≤ onorm f * norm (g x)"
by (rule onorm [OF f])
also have "onorm f * norm (g x) ≤ onorm f * (onorm g * norm x)"
by (rule mult_left_mono [OF onorm [OF g] onorm_pos_le [OF f]])
finally show "norm ((f ∘ g) x) ≤ onorm f * onorm g * norm x"
qed

lemma onorm_scaleR_lemma:
assumes f: "bounded_linear f"
shows "onorm (λx. r *⇩R f x) ≤ ¦r¦ * onorm f"
proof (rule onorm_bound)
show "0 ≤ ¦r¦ * onorm f"
by (intro mult_nonneg_nonneg onorm_pos_le abs_ge_zero f)
next
fix x
have "¦r¦ * norm (f x) ≤ ¦r¦ * (onorm f * norm x)"
by (intro mult_left_mono onorm abs_ge_zero f)
then show "norm (r *⇩R f x) ≤ ¦r¦ * onorm f * norm x"
by (simp only: norm_scaleR mult.assoc)
qed

lemma onorm_scaleR:
assumes f: "bounded_linear f"
shows "onorm (λx. r *⇩R f x) = ¦r¦ * onorm f"
proof (cases "r = 0")
assume "r ≠ 0"
show ?thesis
proof (rule order_antisym)
show "onorm (λx. r *⇩R f x) ≤ ¦r¦ * onorm f"
using f by (rule onorm_scaleR_lemma)
next
have "bounded_linear (λx. r *⇩R f x)"
using bounded_linear_scaleR_right f by (rule bounded_linear_compose)
then have "onorm (λx. inverse r *⇩R r *⇩R f x) ≤ ¦inverse r¦ * onorm (λx. r *⇩R f x)"
by (rule onorm_scaleR_lemma)
with ‹r ≠ 0› show "¦r¦ * onorm f ≤ onorm (λx. r *⇩R f x)"
by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
qed

lemma onorm_scaleR_left_lemma:
assumes r: "bounded_linear r"
shows "onorm (λx. r x *⇩R f) ≤ onorm r * norm f"
proof (rule onorm_bound)
fix x
have "norm (r x *⇩R f) = norm (r x) * norm f"
by simp
also have "… ≤ onorm r * norm x * norm f"
by (intro mult_right_mono onorm r norm_ge_zero)
finally show "norm (r x *⇩R f) ≤ onorm r * norm f * norm x"
qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le r)

lemma onorm_scaleR_left:
assumes f: "bounded_linear r"
shows "onorm (λx. r x *⇩R f) = onorm r * norm f"
proof (cases "f = 0")
assume "f ≠ 0"
show ?thesis
proof (rule order_antisym)
show "onorm (λx. r x *⇩R f) ≤ onorm r * norm f"
using f by (rule onorm_scaleR_left_lemma)
next
have bl1: "bounded_linear (λx. r x *⇩R f)"
by (metis bounded_linear_scaleR_const f)
have "bounded_linear (λx. r x * norm f)"
by (metis bounded_linear_mult_const f)
from onorm_scaleR_left_lemma[OF this, of "inverse (norm f)"]
have "onorm r ≤ onorm (λx. r x * norm f) * inverse (norm f)"
using ‹f ≠ 0›
also have "onorm (λx. r x * norm f) ≤ onorm (λx. r x *⇩R f)"
by (rule onorm_bound)
(auto simp: abs_mult bl1 onorm_pos_le intro!: order_trans[OF _ onorm])
finally show "onorm r * norm f ≤ onorm (λx. r x *⇩R f)"
using ‹f ≠ 0›
by (simp add: inverse_eq_divide pos_le_divide_eq mult.commute)
qed

lemma onorm_neg:
shows "onorm (λx. - f x) = onorm f"
unfolding onorm_def by simp

lemma onorm_triangle:
assumes f: "bounded_linear f"
assumes g: "bounded_linear g"
shows "onorm (λx. f x + g x) ≤ onorm f + onorm g"
proof (rule onorm_bound)
show "0 ≤ onorm f + onorm g"
by (intro add_nonneg_nonneg onorm_pos_le f g)
next
fix x
have "norm (f x + g x) ≤ norm (f x) + norm (g x)"
by (rule norm_triangle_ineq)
also have "norm (f x) + norm (g x) ≤ onorm f * norm x + onorm g * norm x"
by (intro add_mono onorm f g)
finally show "norm (f x + g x) ≤ (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 ≤ e"
shows "onorm (λx. f x + g x) ≤ 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 (λx. f x + g x) < e"
using assms by (rule onorm_triangle [THEN order_le_less_trans])

end
```