Datasets:

hoskinson-center
/

proof-pile

	(*
	File: Banach_Steinhaus.thy
	Author: Dominique Unruh, University of Tartu
	Author: Jose Manuel Rodriguez Caballero, University of Tartu
	*)
	section \<open>Banach-Steinhaus theorem\<close>

	theory Banach_Steinhaus
	imports Banach_Steinhaus_Missing
	begin

	text \<open>
	We formalize Banach-Steinhaus theorem as theorem @{text banach_steinhaus}. This theorem was
	originally proved in Banach-Steinhaus's paper~\cite{banach1927principe}. For the proof, we follow
	Sokal's approach~\cite{sokal2011really}. Furthermore, we prove as a corollary a result about
	pointwise convergent sequences of bounded operators whose domain is a Banach space.
	\<close>

	subsection \<open>Preliminaries for Sokal's proof of Banach-Steinhaus theorem\<close>

	lemma linear_plus_norm:
	includes notation_norm
	assumes \<open>linear f\<close>
	shows \<open>\<parallel>f \<xi>\<parallel> \<le> max \<parallel>f (x + \<xi>)\<parallel> \<parallel>f (x - \<xi>)\<parallel>\<close>
	text \<open>
	Explanation: For arbitrary \<^term>\<open>x\<close> and a linear operator \<^term>\<open>f\<close>,
	\<^term>\<open>norm (f \<xi>)\<close> is upper bounded by the maximum of the norms
	of the shifts of \<^term>\<open>f\<close> (i.e., \<^term>\<open>f (x + \<xi>)\<close> and \<^term>\<open>f (x - \<xi>)\<close>).
	\<close>
	proof-
	have \<open>norm (f \<xi>) = norm ( (inverse (of_nat 2)) *\<^sub>R (f (x + \<xi>) - f (x - \<xi>)) )\<close>
	by (smt add_diff_cancel_left' assms diff_add_cancel diff_diff_add linear_diff midpoint_def
	midpoint_plus_self of_nat_1 of_nat_add one_add_one scaleR_half_double)
	also have \<open>\<dots> = inverse (of_nat 2) * norm (f (x + \<xi>) - f (x - \<xi>))\<close>
	using Real_Vector_Spaces.real_normed_vector_class.norm_scaleR by simp
	also have \<open>\<dots> \<le> inverse (of_nat 2) * (norm (f (x + \<xi>)) + norm (f (x - \<xi>)))\<close>
	by (simp add: norm_triangle_ineq4)
	also have \<open>\<dots> \<le> max (norm (f (x + \<xi>))) (norm (f (x - \<xi>)))\<close>
	by auto
	finally show ?thesis by blast
	qed

	lemma onorm_Sup_on_ball:
	includes notation_norm
	assumes \<open>r > 0\<close>
	shows "\<parallel>f\<parallel> \<le> Sup ( (\<lambda>x. \<parallel>f *\<^sub>v x\<parallel>) ` (ball x r) ) / r"
	text \<open>
	Explanation: Let \<^term>\<open>f\<close> be a bounded operator and let \<^term>\<open>x\<close> be a point. For any \<^term>\<open>r > 0\<close>,
	the operator norm of \<^term>\<open>f\<close> is bounded above by the supremum of $f$ applied to the open ball of
	radius \<^term>\<open>r\<close> around \<^term>\<open>x\<close>, divided by \<^term>\<open>r\<close>.
	\<close>
	proof-
	have bdd_above_3: \<open>bdd_above ((\<lambda>x. \<parallel>f *\<^sub>v x\<parallel>) ` (ball 0 r))\<close>
	proof -
	obtain M where \<open>\<And> \<xi>. \<parallel>f \<^sub>v \<xi>\<parallel> \<le> M norm \<xi>\<close> and \<open>M \<ge> 0\<close>
	using norm_blinfun norm_ge_zero by blast
	hence \<open>\<And> \<xi>. \<xi> \<in> ball 0 r \<Longrightarrow> \<parallel>f \<^sub>v \<xi>\<parallel> \<le> M r\<close>
	using \<open>r > 0\<close> by (smt mem_ball_0 mult_left_mono)
	thus ?thesis by (meson bdd_aboveI2)
	qed
	have bdd_above_2: \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r))\<close>
	proof-
	have \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v x\<parallel>) ` (ball 0 r))\<close>
	by auto
	moreover have \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v \<xi>\<parallel>) ` (ball 0 r))\<close>
	using bdd_above_3 by blast
	ultimately have \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f \<^sub>v x\<parallel> + \<parallel>f \<^sub>v \<xi>\<parallel>) ` (ball 0 r))\<close>
	by (rule bdd_above_plus)
	then obtain M where \<open>\<And> \<xi>. \<xi> \<in> ball 0 r \<Longrightarrow> \<parallel>f \<^sub>v x\<parallel> + \<parallel>f \<^sub>v \<xi>\<parallel> \<le> M\<close>
	unfolding bdd_above_def by (meson image_eqI)
	moreover have \<open>\<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<le> \<parallel>f \<^sub>v x\<parallel> + \<parallel>f *\<^sub>v \<xi>\<parallel>\<close> for \<xi>
	by (simp add: blinfun.add_right norm_triangle_ineq)
	ultimately have \<open>\<And> \<xi>. \<xi> \<in> ball 0 r \<Longrightarrow> \<parallel>f *\<^sub>v (x + \<xi>)\<parallel> \<le> M\<close>
	by (simp add: blinfun.add_right norm_triangle_le)
	thus ?thesis by (meson bdd_aboveI2)
	qed
	have bdd_above_4: \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	proof-
	obtain K where K_def: \<open>\<And> \<xi>. \<xi> \<in> ball 0 r \<Longrightarrow> \<parallel>f *\<^sub>v (x + \<xi>)\<parallel> \<le> K\<close>
	using \<open>bdd_above ((\<lambda> \<xi>. norm (f (x + \<xi>))) ` (ball 0 r))\<close> unfolding bdd_above_def
	by (meson image_eqI)
	have \<open>\<xi> \<in> ball (0::'a) r \<Longrightarrow> -\<xi> \<in> ball 0 r\<close> for \<xi>
	by auto
	thus ?thesis by (metis K_def ab_group_add_class.ab_diff_conv_add_uminus bdd_aboveI2)
	qed
	have bdd_above_1: \<open>bdd_above ((\<lambda> \<xi>. max \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	proof-
	have \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r))\<close>
	using bdd_above_2 by blast
	moreover have \<open>bdd_above ((\<lambda> \<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	using bdd_above_4 by blast
	ultimately show ?thesis
	unfolding max_def apply auto apply (meson bdd_above_Int1 bdd_above_mono image_Int_subset)
	by (meson bdd_above_Int1 bdd_above_mono image_Int_subset)
	qed
	have bdd_above_6: \<open>bdd_above ((\<lambda>t. \<parallel>f *\<^sub>v t\<parallel>) ` ball x r)\<close>
	proof-
	have \<open>bounded (ball x r)\<close>
	by simp
	hence \<open>bounded ((\<lambda>t. \<parallel>f *\<^sub>v t\<parallel>) ` ball x r)\<close>
	by (metis (no_types) add.left_neutral bdd_above_2 bdd_above_norm bounded_norm_comp
	image_add_ball image_image)
	thus ?thesis
	by (simp add: bounded_imp_bdd_above)
	qed
	have norm_1: \<open>(\<lambda>\<xi>. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) ` ball 0 r = (\<lambda>t. \<parallel>f \<^sub>v t\<parallel>) ` ball x r\<close>
	by (metis add.right_neutral ball_translation image_image)
	have bdd_above_5: \<open>bdd_above ((\<lambda>\<xi>. norm (f (x + \<xi>))) ` ball 0 r)\<close>
	by (simp add: bdd_above_2)
	have norm_2: \<open>\<parallel>\<xi>\<parallel> < r \<Longrightarrow> \<parallel>f \<^sub>v (x - \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) ` ball 0 r\<close> for \<xi>
	proof-
	assume \<open>\<parallel>\<xi>\<parallel> < r\<close>
	hence \<open>\<xi> \<in> ball (0::'a) r\<close>
	by auto
	hence \<open>-\<xi> \<in> ball (0::'a) r\<close>
	by auto
	thus ?thesis
	by (metis (no_types, lifting) ab_group_add_class.ab_diff_conv_add_uminus image_iff)
	qed
	have norm_2': \<open>\<parallel>\<xi>\<parallel> < r \<Longrightarrow> \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` ball 0 r\<close> for \<xi>
	proof-
	assume \<open>norm \<xi> < r\<close>
	hence \<open>\<xi> \<in> ball (0::'a) r\<close>
	by auto
	hence \<open>-\<xi> \<in> ball (0::'a) r\<close>
	by auto
	thus ?thesis
	by (metis (no_types, lifting) diff_minus_eq_add image_iff)
	qed
	have bdd_above_6: \<open>bdd_above ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` ball 0 r)\<close>
	by (simp add: bdd_above_4)
	have Sup_2: \<open>(SUP \<xi>\<in>ball 0 r. max \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) =
	max (SUP \<xi>\<in>ball 0 r. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) (SUP \<xi>\<in>ball 0 r. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>)\<close>
	proof-
	have \<open>ball (0::'a) r \<noteq> {}\<close>
	using \<open>r > 0\<close> by auto
	moreover have \<open>bdd_above ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` ball 0 r)\<close>
	using bdd_above_5 by blast
	moreover have \<open>bdd_above ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` ball 0 r)\<close>
	using bdd_above_6 by blast
	ultimately show ?thesis
	using max_Sup
	by (metis (mono_tags, lifting) Banach_Steinhaus_Missing.pointwise_max_def image_cong)
	qed
	have Sup_3': \<open>\<parallel>\<xi>\<parallel> < r \<Longrightarrow> \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` ball 0 r\<close> for \<xi>::'a
	by (simp add: norm_2')
	have Sup_3'': \<open>\<parallel>\<xi>\<parallel> < r \<Longrightarrow> \<parallel>f \<^sub>v (x - \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) ` ball 0 r\<close> for \<xi>::'a
	by (simp add: norm_2)
	have Sup_3: \<open>max (SUP \<xi>\<in>ball 0 r. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) (SUP \<xi>\<in>ball 0 r. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) =
	(SUP \<xi>\<in>ball 0 r. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>)\<close>
	proof-
	have \<open>(\<lambda>\<xi>. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r) = (\<lambda>\<xi>. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r)\<close>
	apply auto using Sup_3' apply auto using Sup_3'' by blast
	hence \<open>Sup ((\<lambda>\<xi>. \<parallel>f \<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r))=Sup ((\<lambda>\<xi>. \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	by simp
	thus ?thesis by simp
	qed
	have Sup_1: \<open>Sup ((\<lambda>t. \<parallel>f \<^sub>v t\<parallel>) ` (ball 0 r)) \<le> Sup ( (\<lambda>\<xi>. \<parallel>f \<^sub>v \<xi>\<parallel>) ` (ball x r) )\<close>
	proof-
	have \<open>(\<lambda>t. \<parallel>f \<^sub>v t\<parallel>) \<xi> \<le> max \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>\<close> for \<xi>
	apply(rule linear_plus_norm) apply (rule bounded_linear.linear)
	by (simp add: blinfun.bounded_linear_right)
	moreover have \<open>bdd_above ((\<lambda> \<xi>. max \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	using bdd_above_1 by blast
	moreover have \<open>ball (0::'a) r \<noteq> {}\<close>
	using \<open>r > 0\<close> by auto
	ultimately have \<open>Sup ((\<lambda>t. \<parallel>f *\<^sub>v t\<parallel>) ` (ball 0 r)) \<le>
	Sup ((\<lambda>\<xi>. max \<parallel>f \<^sub>v (x + \<xi>)\<parallel> \<parallel>f \<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r))\<close>
	using cSUP_mono by smt
	also have \<open>\<dots> = max (Sup ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r)))
	(Sup ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x - \<xi>)\<parallel>) ` (ball 0 r)))\<close>
	using Sup_2 by blast
	also have \<open>\<dots> = Sup ((\<lambda>\<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` (ball 0 r))\<close>
	using Sup_3 by blast
	also have \<open>\<dots> = Sup ((\<lambda>\<xi>. \<parallel>f *\<^sub>v \<xi>\<parallel>) ` (ball x r))\<close>
	by (metis add.right_neutral ball_translation image_image)
	finally show ?thesis by blast
	qed
	have \<open>\<parallel>f\<parallel> = (SUP x\<in>ball 0 r. \<parallel>f *\<^sub>v x\<parallel>) / r\<close>
	using \<open>0 < r\<close> onorm_r by blast
	moreover have \<open>Sup ((\<lambda>t. \<parallel>f \<^sub>v t\<parallel>) ` (ball 0 r)) / r \<le> Sup ((\<lambda>\<xi>. \<parallel>f \<^sub>v \<xi>\<parallel>) ` (ball x r)) / r\<close>
	using Sup_1 \<open>0 < r\<close> divide_right_mono by fastforce
	ultimately have \<open>\<parallel>f\<parallel> \<le> Sup ((\<lambda>t. \<parallel>f *\<^sub>v t\<parallel>) ` ball x r) / r\<close>
	by simp
	thus ?thesis by simp
	qed

	lemma onorm_Sup_on_ball':
	includes notation_norm
	assumes \<open>r > 0\<close> and \<open>\<tau> < 1\<close>
	shows \<open>\<exists>\<xi>\<in>ball x r. \<tau> * r * \<parallel>f\<parallel> \<le> \<parallel>f *\<^sub>v \<xi>\<parallel>\<close>
	text \<open>
	In the proof of Banach-Steinhaus theorem, we will use this variation of the
	lemma @{text onorm_Sup_on_ball}.

	Explanation: Let \<^term>\<open>f\<close> be a bounded operator, let \<^term>\<open>x\<close> be a point and let \<^term>\<open>r\<close> be a
	positive real number. For any real number \<^term>\<open>\<tau> < 1\<close>, there is a point \<^term>\<open>\<xi>\<close> in the open ball
	of radius \<^term>\<open>r\<close> around \<^term>\<open>x\<close> such that \<^term>\<open>\<tau> * r * \<parallel>f\<parallel> \<le> \<parallel>f *\<^sub>v \<xi>\<parallel>\<close>.
	\<close>
	proof(cases \<open>f = 0\<close>)
	case True
	thus ?thesis by (metis assms(1) centre_in_ball mult_zero_right norm_zero order_refl
	zero_blinfun.rep_eq)
	next
	case False
	have bdd_above_1: \<open>bdd_above ((\<lambda>t. \<parallel>(*\<^sub>v) f t\<parallel>) ` ball x r)\<close> for f::\<open>'a \<Rightarrow>\<^sub>L 'b\<close>
	using assms(1) bounded_linear_image by (simp add: bounded_linear_image
	blinfun.bounded_linear_right bounded_imp_bdd_above bounded_norm_comp)
	have \<open>norm f > 0\<close>
	using \<open>f \<noteq> 0\<close> by auto
	have \<open>norm f \<le> Sup ( (\<lambda>\<xi>. \<parallel>(*\<^sub>v) f \<xi>\<parallel>) ` (ball x r) ) / r\<close>
	using \<open>r > 0\<close> by (simp add: onorm_Sup_on_ball)
	hence \<open>r * norm f \<le> Sup ( (\<lambda>\<xi>. \<parallel>(*\<^sub>v) f \<xi>\<parallel>) ` (ball x r) )\<close>
	using \<open>0 < r\<close> by (smt divide_strict_right_mono nonzero_mult_div_cancel_left)
	moreover have \<open>\<tau> * r * norm f < r * norm f\<close>
	using \<open>\<tau> < 1\<close> using \<open>0 < norm f\<close> \<open>0 < r\<close> by auto
	ultimately have \<open>\<tau> * r * norm f < Sup ( (norm \<circ> ((*\<^sub>v) f)) ` (ball x r) )\<close>
	by simp
	moreover have \<open>(norm \<circ> ( (*\<^sub>v) f)) ` (ball x r) \<noteq> {}\<close>
	using \<open>0 < r\<close> by auto
	moreover have \<open>bdd_above ((norm \<circ> ( (*\<^sub>v) f)) ` (ball x r))\<close>
	using bdd_above_1 apply transfer by simp
	ultimately have \<open>\<exists>t \<in> (norm \<circ> ( (\<^sub>v) f)) ` (ball x r). \<tau> r * norm f < t\<close>
	by (simp add: less_cSup_iff)
	thus ?thesis by (smt comp_def image_iff)
	qed

	subsection \<open>Banach-Steinhaus theorem\<close>

	theorem banach_steinhaus:
	fixes f::\<open>'c \<Rightarrow> ('a::banach \<Rightarrow>\<^sub>L 'b::real_normed_vector)\<close>
	assumes \<open>\<And>x. bounded (range (\<lambda>n. (f n) *\<^sub>v x))\<close>
	shows \<open>bounded (range f)\<close>
	text\<open>
	This is Banach-Steinhaus Theorem.

	Explanation: If a family of bounded operators on a Banach space
	is pointwise bounded, then it is uniformly bounded.
	\<close>
	proof(rule classical)
	assume \<open>\<not>(bounded (range f))\<close>
	have sum_1: \<open>\<exists>K. \<forall>n. sum (\<lambda>k. inverse (real_of_nat 3^k)) {0..n} \<le> K\<close>
	proof-
	have \<open>summable (\<lambda>n. inverse ((3::real) ^ n))\<close>
	by (simp flip: power_inverse)
	hence \<open>bounded (range (\<lambda>n. sum (\<lambda> k. inverse (real 3 ^ k)) {0..<n}))\<close>
	using summable_imp_sums_bounded[where f = "(\<lambda>n. inverse (real_of_nat 3^n))"]
	lessThan_atLeast0 by auto
	hence \<open>\<exists>M. \<forall>h\<in>(range (\<lambda>n. sum (\<lambda> k. inverse (real 3 ^ k)) {0..<n})). norm h \<le> M\<close>
	using bounded_iff by blast
	then obtain M where \<open>h\<in>range (\<lambda>n. sum (\<lambda> k. inverse (real 3 ^ k)) {0..<n}) \<Longrightarrow> norm h \<le> M\<close>
	for h
	by blast
	have sum_2: \<open>sum (\<lambda>k. inverse (real_of_nat 3^k)) {0..n} \<le> M\<close> for n
	proof-
	have \<open>norm (sum (\<lambda> k. inverse (real 3 ^ k)) {0..< Suc n}) \<le> M\<close>
	using \<open>\<And>h. h\<in>(range (\<lambda>n. sum (\<lambda> k. inverse (real 3 ^ k)) {0..<n})) \<Longrightarrow> norm h \<le> M\<close>
	by blast
	hence \<open>norm (sum (\<lambda> k. inverse (real 3 ^ k)) {0..n}) \<le> M\<close>
	by (simp add: atLeastLessThanSuc_atLeastAtMost)
	hence \<open>(sum (\<lambda> k. inverse (real 3 ^ k)) {0..n}) \<le> M\<close>
	by auto
	thus ?thesis by blast
	qed
	have \<open>sum (\<lambda>k. inverse (real_of_nat 3^k)) {0..n} \<le> M\<close> for n
	using sum_2 by blast
	thus ?thesis by blast
	qed
	have \<open>of_rat 2/3 < (1::real)\<close>
	by auto
	hence \<open>\<forall>g::'a \<Rightarrow>\<^sub>L 'b. \<forall>x. \<forall>r. \<exists>\<xi>. g \<noteq> 0 \<and> r > 0
	\<longrightarrow> (\<xi>\<in>ball x r \<and> (of_rat 2/3) * r * norm g \<le> norm ((*\<^sub>v) g \<xi>))\<close>
	using onorm_Sup_on_ball' by blast
	hence \<open>\<exists>\<xi>. \<forall>g::'a \<Rightarrow>\<^sub>L 'b. \<forall>x. \<forall>r. g \<noteq> 0 \<and> r > 0
	\<longrightarrow> ((\<xi> g x r)\<in>ball x r \<and> (of_rat 2/3) * r * norm g \<le> norm ((*\<^sub>v) g (\<xi> g x r)))\<close>
	by metis
	then obtain \<xi> where f1: \<open>\<lbrakk>g \<noteq> 0; r > 0\<rbrakk> \<Longrightarrow>
	\<xi> g x r \<in> ball x r \<and> (of_rat 2/3) * r * norm g \<le> norm ((*\<^sub>v) g (\<xi> g x r))\<close>
	for g::\<open>'a \<Rightarrow>\<^sub>L 'b\<close> and x and r
	by blast
	have \<open>\<forall>n. \<exists>k. norm (f k) \<ge> 4^n\<close>
	using \<open>\<not>(bounded (range f))\<close> by (metis (mono_tags, opaque_lifting) boundedI image_iff linear)
	hence \<open>\<exists>k. \<forall>n. norm (f (k n)) \<ge> 4^n\<close>
	by metis
	hence \<open>\<exists>k. \<forall>n. norm ((f \<circ> k) n) \<ge> 4^n\<close>
	by simp
	then obtain k where \<open>norm ((f \<circ> k) n) \<ge> 4^n\<close> for n
	by blast
	define T where \<open>T = f \<circ> k\<close>
	have \<open>T n \<in> range f\<close> for n
	unfolding T_def by simp
	have \<open>norm (T n) \<ge> of_nat (4^n)\<close> for n
	unfolding T_def using \<open>\<And> n. norm ((f \<circ> k) n) \<ge> 4^n\<close> by auto
	hence \<open>T n \<noteq> 0\<close> for n
	by (smt T_def \<open>\<And>n. 4 ^ n \<le> norm ((f \<circ> k) n)\<close> norm_zero power_not_zero zero_le_power)
	have \<open>inverse (of_nat 3^n) > (0::real)\<close> for n
	by auto
	define y::\<open>nat \<Rightarrow> 'a\<close> where \<open>y = rec_nat 0 (\<lambda>n x. \<xi> (T n) x (inverse (of_nat 3^n)))\<close>
	have \<open>y (Suc n) \<in> ball (y n) (inverse (of_nat 3^n))\<close> for n
	using f1 \<open>\<And> n. T n \<noteq> 0\<close> \<open>\<And> n. inverse (of_nat 3^n) > 0\<close> unfolding y_def by auto
	hence \<open>norm (y (Suc n) - y n) \<le> inverse (of_nat 3^n)\<close> for n
	unfolding ball_def apply auto using dist_norm by (smt norm_minus_commute)
	moreover have \<open>\<exists>K. \<forall>n. sum (\<lambda>k. inverse (real_of_nat 3^k)) {0..n} \<le> K\<close>
	using sum_1 by blast
	moreover have \<open>Cauchy y\<close>
	using convergent_series_Cauchy[where a = "\<lambda>n. inverse (of_nat 3^n)" and \<phi> = y] dist_norm
	by (metis calculation(1) calculation(2))
	hence \<open>\<exists> x. y \<longlonglongrightarrow> x\<close>
	by (simp add: convergent_eq_Cauchy)
	then obtain x where \<open>y \<longlonglongrightarrow> x\<close>
	by blast
	have norm_2: \<open>norm (x - y (Suc n)) \<le> (inverse (of_nat 2))*(inverse (of_nat 3^n))\<close> for n
	proof-
	have \<open>inverse (real_of_nat 3) < 1\<close>
	by simp
	moreover have \<open>y 0 = 0\<close>
	using y_def by auto
	ultimately have \<open>norm (x - y (Suc n))
	\<le> (inverse (of_nat 3)) * inverse (1 - (inverse (of_nat 3))) * ((inverse (of_nat 3)) ^ n)\<close>
	using bound_Cauchy_to_lim[where c = "inverse (of_nat 3)" and y = y and x = x]
	power_inverse semiring_norm(77) \<open>y \<longlonglongrightarrow> x\<close>
	\<open>\<And> n. norm (y (Suc n) - y n) \<le> inverse (of_nat 3^n)\<close> by (metis divide_inverse)
	moreover have \<open>inverse (real_of_nat 3) * inverse (1 - (inverse (of_nat 3)))
	= inverse (of_nat 2)\<close>
	by auto
	ultimately show ?thesis
	by (metis power_inverse)
	qed
	have \<open>norm (x - y (Suc n)) \<le> (inverse (of_nat 2))*(inverse (of_nat 3^n))\<close> for n
	using norm_2 by blast
	have \<open>\<exists> M. \<forall> n. norm ((*\<^sub>v) (T n) x) \<le> M\<close>
	unfolding T_def apply auto
	by (metis \<open>\<And>x. bounded (range (\<lambda>n. (*\<^sub>v) (f n) x))\<close> bounded_iff rangeI)
	then obtain M where \<open>norm ((*\<^sub>v) (T n) x) \<le> M\<close> for n
	by blast
	have norm_1: \<open>norm (T n) * norm (y (Suc n) - x) + norm ((*\<^sub>v) (T n) x)
	\<le> inverse (real 2) * inverse (real 3 ^ n) * norm (T n) + norm ((*\<^sub>v) (T n) x)\<close> for n
	proof-
	have \<open>norm (y (Suc n) - x) \<le> (inverse (of_nat 2))*(inverse (of_nat 3^n))\<close>
	using \<open>norm (x - y (Suc n)) \<le> (inverse (of_nat 2))*(inverse (of_nat 3^n))\<close>
	by (simp add: norm_minus_commute)
	moreover have \<open>norm (T n) \<ge> 0\<close>
	by auto
	ultimately have \<open>norm (T n) * norm (y (Suc n) - x)
	\<le> (inverse (of_nat 2))(inverse (of_nat 3^n))norm (T n)\<close>
	by (simp add: \<open>\<And>n. T n \<noteq> 0\<close>)
	thus ?thesis by simp
	qed
	have inverse_2: \<open>(inverse (of_nat 6)) * inverse (real 3 ^ n) * norm (T n)
	\<le> norm ((*\<^sub>v) (T n) x)\<close> for n
	proof-
	have \<open>(of_rat 2/3)(inverse (of_nat 3^n))norm (T n) \<le> norm ((*\<^sub>v) (T n) (y (Suc n)))\<close>
	using f1 \<open>\<And> n. T n \<noteq> 0\<close> \<open>\<And> n. inverse (of_nat 3^n) > 0\<close> unfolding y_def by auto
	also have \<open>\<dots> = norm ((*\<^sub>v) (T n) ((y (Suc n) - x) + x))\<close>
	by auto
	also have \<open>\<dots> = norm ((\<^sub>v) (T n) (y (Suc n) - x) + (\<^sub>v) (T n) x)\<close>
	apply transfer apply auto by (metis diff_add_cancel linear_simps(1))
	also have \<open>\<dots> \<le> norm ((\<^sub>v) (T n) (y (Suc n) - x)) + norm ((\<^sub>v) (T n) x)\<close>
	by (simp add: norm_triangle_ineq)
	also have \<open>\<dots> \<le> norm (T n) * norm (y (Suc n) - x) + norm ((*\<^sub>v) (T n) x)\<close>
	apply transfer apply auto using onorm by auto
	also have \<open>\<dots> \<le> (inverse (of_nat 2))(inverse (of_nat 3^n))norm (T n)
	+ norm ((*\<^sub>v) (T n) x)\<close>
	using norm_1 by blast
	finally have \<open>(of_rat 2/3) * inverse (real 3 ^ n) * norm (T n)
	\<le> inverse (real 2) * inverse (real 3 ^ n) * norm (T n)
	+ norm ((*\<^sub>v) (T n) x)\<close>
	by blast
	hence \<open>(of_rat 2/3) * inverse (real 3 ^ n) * norm (T n)
	- inverse (real 2) * inverse (real 3 ^ n) * norm (T n) \<le> norm ((*\<^sub>v) (T n) x)\<close>
	by linarith
	moreover have \<open>(of_rat 2/3) * inverse (real 3 ^ n) * norm (T n)
	- inverse (real 2) * inverse (real 3 ^ n) * norm (T n)
	= (inverse (of_nat 6)) * inverse (real 3 ^ n) * norm (T n)\<close>
	by fastforce
	ultimately show \<open>(inverse (of_nat 6)) * inverse (real 3 ^ n) * norm (T n) \<le> norm ((*\<^sub>v) (T n) x)\<close>
	by linarith
	qed
	have inverse_3: \<open>(inverse (of_nat 6)) * (of_rat (4/3)^n)
	\<le> (inverse (of_nat 6)) * inverse (real 3 ^ n) * norm (T n)\<close> for n
	proof-
	have \<open>of_rat (4/3)^n = inverse (real 3 ^ n) * (of_nat 4^n)\<close>
	apply auto by (metis divide_inverse_commute of_rat_divide power_divide of_rat_numeral_eq)
	also have \<open>\<dots> \<le> inverse (real 3 ^ n) * norm (T n)\<close>
	using \<open>\<And>n. norm (T n) \<ge> of_nat (4^n)\<close> by simp
	finally have \<open>of_rat (4/3)^n \<le> inverse (real 3 ^ n) * norm (T n)\<close>
	by blast
	moreover have \<open>inverse (of_nat 6) > (0::real)\<close>
	by auto
	ultimately show ?thesis by auto
	qed
	have inverse_1: \<open>(inverse (of_nat 6)) * (of_rat (4/3)^n) \<le> M\<close> for n
	proof-
	have \<open>(inverse (of_nat 6)) * (of_rat (4/3)^n)
	\<le> (inverse (of_nat 6)) * inverse (real 3 ^ n) * norm (T n)\<close>
	using inverse_3 by blast
	also have \<open>\<dots> \<le> norm ((*\<^sub>v) (T n) x)\<close>
	using inverse_2 by blast
	finally have \<open>(inverse (of_nat 6)) * (of_rat (4/3)^n) \<le> norm ((*\<^sub>v) (T n) x)\<close>
	by auto
	thus ?thesis using \<open>\<And> n. norm ((*\<^sub>v) (T n) x) \<le> M\<close> by smt
	qed
	have \<open>\<exists>n. M < (inverse (of_nat 6)) * (of_rat (4/3)^n)\<close>
	using Real.real_arch_pow by auto
	moreover have \<open>(inverse (of_nat 6)) * (of_rat (4/3)^n) \<le> M\<close> for n
	using inverse_1 by blast
	ultimately show ?thesis by smt
	qed

	subsection \<open>A consequence of Banach-Steinhaus theorem\<close>

	corollary bounded_linear_limit_bounded_linear:
	fixes f::\<open>nat \<Rightarrow> ('a::banach \<Rightarrow>\<^sub>L 'b::real_normed_vector)\<close>
	assumes \<open>\<And>x. convergent (\<lambda>n. (f n) *\<^sub>v x)\<close>
	shows \<open>\<exists>g. (\<lambda>n. (\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> (\<^sub>v) g\<close>
	text\<open>
	Explanation: If a sequence of bounded operators on a Banach space converges
	pointwise, then the limit is also a bounded operator.
	\<close>
	proof-
	have \<open>\<exists>l. (\<lambda>n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> l\<close> for x
	by (simp add: \<open>\<And>x. convergent (\<lambda>n. (*\<^sub>v) (f n) x)\<close> convergentD)
	hence \<open>\<exists>F. (\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close>
	unfolding pointwise_convergent_to_def by metis
	obtain F where \<open>(\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close>
	using \<open>\<exists>F. (\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close> by auto
	have \<open>\<And>x. (\<lambda> n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close>
	using \<open>(\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close> apply transfer
	by (simp add: pointwise_convergent_to_def)
	have \<open>bounded (range f)\<close>
	using \<open>\<And>x. convergent (\<lambda>n. (*\<^sub>v) (f n) x)\<close> banach_steinhaus
	\<open>\<And>x. \<exists>l. (\<lambda>n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> l\<close> convergent_imp_bounded by blast
	have norm_f_n: \<open>\<exists> M. \<forall> n. norm (f n) \<le> M\<close>
	unfolding bounded_def
	by (meson UNIV_I \<open>bounded (range f)\<close> bounded_iff image_eqI)
	have \<open>isCont (\<lambda> t::'b. norm t) y\<close> for y::'b
	using Limits.isCont_norm by simp
	hence \<open>(\<lambda> n. norm ((*\<^sub>v) (f n) x)) \<longlonglongrightarrow> (norm (F x))\<close> for x
	using \<open>\<And> x::'a. (\<lambda> n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> by (simp add: tendsto_norm)
	hence norm_f_n_x: \<open>\<exists> M. \<forall> n. norm ((*\<^sub>v) (f n) x) \<le> M\<close> for x
	using Elementary_Metric_Spaces.convergent_imp_bounded
	by (metis UNIV_I \<open>\<And> x::'a. (\<lambda> n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> bounded_iff image_eqI)
	have norm_f: \<open>\<exists>K. \<forall>n. \<forall>x. norm ((\<^sub>v) (f n) x) \<le> norm x K\<close>
	proof-
	have \<open>\<exists> M. \<forall> n. norm ((*\<^sub>v) (f n) x) \<le> M\<close> for x
	using norm_f_n_x \<open>\<And>x. (\<lambda>n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> by blast
	hence \<open>\<exists> M. \<forall> n. norm (f n) \<le> M\<close>
	using norm_f_n by simp
	then obtain M::real where \<open>\<exists> M. \<forall> n. norm (f n) \<le> M\<close>
	by blast
	have \<open>\<forall> n. \<forall>x. norm ((\<^sub>v) (f n) x) \<le> norm x norm (f n)\<close>
	apply transfer apply auto by (metis mult.commute onorm)
	thus ?thesis using \<open>\<exists> M. \<forall> n. norm (f n) \<le> M\<close>
	by (metis (no_types, opaque_lifting) dual_order.trans norm_eq_zero order_refl
	mult_le_cancel_iff2 vector_space_over_itself.scale_zero_left zero_less_norm_iff)
	qed
	have norm_F_x: \<open>\<exists>K. \<forall>x. norm (F x) \<le> norm x * K\<close>
	proof-
	have "\<exists>K. \<forall>n. \<forall>x. norm ((\<^sub>v) (f n) x) \<le> norm x K"
	using norm_f \<open>\<And>x. (\<lambda>n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> by auto
	thus ?thesis
	using \<open>\<And> x::'a. (\<lambda> n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> apply transfer
	by (metis Lim_bounded tendsto_norm)
	qed
	have \<open>linear F\<close>
	proof(rule linear_limit_linear)
	show \<open>linear ((*\<^sub>v) (f n))\<close> for n
	apply transfer apply auto by (simp add: bounded_linear.linear)
	show \<open>f \<midarrow>pointwise\<rightarrow> F\<close>
	using \<open>(\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close> by auto
	qed
	moreover have \<open>bounded_linear_axioms F\<close>
	using norm_F_x by (simp add: \<open>\<And>x. (\<lambda>n. (*\<^sub>v) (f n) x) \<longlonglongrightarrow> F x\<close> bounded_linear_axioms_def)
	ultimately have \<open>bounded_linear F\<close>
	unfolding bounded_linear_def by blast
	hence \<open>\<exists>g. (*\<^sub>v) g = F\<close>
	using bounded_linear_Blinfun_apply by auto
	thus ?thesis using \<open>(\<lambda>n. (*\<^sub>v) (f n)) \<midarrow>pointwise\<rightarrow> F\<close> apply transfer by auto
	qed

	end