src/HOL/Analysis/Bounded_Linear_Function.thy

tuned

(*  Title:      HOL/Analysis/Bounded_Linear_Function.thy
    Author:     Fabian Immler, TU München
*)

section \<open>Bounded Linear Function\<close>

theory Bounded_Linear_Function
imports
  Topology_Euclidean_Space
  Operator_Norm
begin

subsection \<open>Intro rules for @{term bounded_linear}\<close>

named_theorems bounded_linear_intros

lemma onorm_inner_left:
  assumes "bounded_linear r"
  shows "onorm (\<lambda>x. r x \<bullet> f) \<le> onorm r * norm f"
proof (rule onorm_bound)
  fix x
  have "norm (r x \<bullet> f) \<le> norm (r x) * norm f"
    by (simp add: Cauchy_Schwarz_ineq2)
  also have "\<dots> \<le> onorm r * norm x * norm f"
    by (intro mult_right_mono onorm assms norm_ge_zero)
  finally show "norm (r x \<bullet> f) \<le> onorm r * norm f * norm x"
    by (simp add: ac_simps)
qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms)

lemma onorm_inner_right:
  assumes "bounded_linear r"
  shows "onorm (\<lambda>x. f \<bullet> r x) \<le> norm f * onorm r"
  apply (subst inner_commute)
  apply (rule onorm_inner_left[OF assms, THEN order_trans])
  apply simp
  done

lemmas [bounded_linear_intros] =
  bounded_linear_zero
  bounded_linear_add
  bounded_linear_const_mult
  bounded_linear_mult_const
  bounded_linear_scaleR_const
  bounded_linear_const_scaleR
  bounded_linear_ident
  bounded_linear_sum
  bounded_linear_Pair
  bounded_linear_sub
  bounded_linear_fst_comp
  bounded_linear_snd_comp
  bounded_linear_inner_left_comp
  bounded_linear_inner_right_comp


subsection \<open>declaration of derivative/continuous/tendsto introduction rules for bounded linear functions\<close>

attribute_setup bounded_linear =
  \<open>Scan.succeed (Thm.declaration_attribute (fn thm =>
    fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
      [
        (@{thm bounded_linear.has_derivative}, @{named_theorems derivative_intros}),
        (@{thm bounded_linear.tendsto}, @{named_theorems tendsto_intros}),
        (@{thm bounded_linear.continuous}, @{named_theorems continuous_intros}),
        (@{thm bounded_linear.continuous_on}, @{named_theorems continuous_intros}),
        (@{thm bounded_linear.uniformly_continuous_on}, @{named_theorems continuous_intros}),
        (@{thm bounded_linear_compose}, @{named_theorems bounded_linear_intros})
      ]))\<close>

attribute_setup bounded_bilinear =
  \<open>Scan.succeed (Thm.declaration_attribute (fn thm =>
    fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
      [
        (@{thm bounded_bilinear.FDERIV}, @{named_theorems derivative_intros}),
        (@{thm bounded_bilinear.tendsto}, @{named_theorems tendsto_intros}),
        (@{thm bounded_bilinear.continuous}, @{named_theorems continuous_intros}),
        (@{thm bounded_bilinear.continuous_on}, @{named_theorems continuous_intros}),
        (@{thm bounded_linear_compose[OF bounded_bilinear.bounded_linear_left]},
          @{named_theorems bounded_linear_intros}),
        (@{thm bounded_linear_compose[OF bounded_bilinear.bounded_linear_right]},
          @{named_theorems bounded_linear_intros}),
        (@{thm bounded_linear.uniformly_continuous_on[OF bounded_bilinear.bounded_linear_left]},
          @{named_theorems continuous_intros}),
        (@{thm bounded_linear.uniformly_continuous_on[OF bounded_bilinear.bounded_linear_right]},
          @{named_theorems continuous_intros})
      ]))\<close>


subsection \<open>type of bounded linear functions\<close>

typedef (overloaded) ('a, 'b) blinfun ("(_ \<Rightarrow>\<^sub>L /_)" [22, 21] 21) =
  "{f::'a::real_normed_vector\<Rightarrow>'b::real_normed_vector. bounded_linear f}"
  morphisms blinfun_apply Blinfun
  by (blast intro: bounded_linear_intros)

declare [[coercion
    "blinfun_apply :: ('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'b"]]

lemma bounded_linear_blinfun_apply[bounded_linear_intros]:
  "bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. blinfun_apply f (g x))"
  by (metis blinfun_apply mem_Collect_eq bounded_linear_compose)

setup_lifting type_definition_blinfun

lemma blinfun_eqI: "(\<And>i. blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
  by transfer auto

lemma bounded_linear_Blinfun_apply: "bounded_linear f \<Longrightarrow> blinfun_apply (Blinfun f) = f"
  by (auto simp: Blinfun_inverse)


subsection \<open>type class instantiations\<close>

instantiation blinfun :: (real_normed_vector, real_normed_vector) real_normed_vector
begin

lift_definition norm_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> real" is onorm .

lift_definition minus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  is "\<lambda>f g x. f x - g x"
  by (rule bounded_linear_sub)

definition dist_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> real"
  where "dist_blinfun a b = norm (a - b)"

definition [code del]:
  "(uniformity :: (('a \<Rightarrow>\<^sub>L 'b) \<times> ('a \<Rightarrow>\<^sub>L 'b)) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

definition open_blinfun :: "('a \<Rightarrow>\<^sub>L 'b) set \<Rightarrow> bool"
  where [code del]: "open_blinfun S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"

lift_definition uminus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" is "\<lambda>f x. - f x"
  by (rule bounded_linear_minus)

lift_definition zero_blinfun :: "'a \<Rightarrow>\<^sub>L 'b" is "\<lambda>x. 0"
  by (rule bounded_linear_zero)

lift_definition plus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  is "\<lambda>f g x. f x + g x"
  by (metis bounded_linear_add)

lift_definition scaleR_blinfun::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" is "\<lambda>r f x. r *\<^sub>R f x"
  by (metis bounded_linear_compose bounded_linear_scaleR_right)

definition sgn_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  where "sgn_blinfun x = scaleR (inverse (norm x)) x"

instance
  apply standard
  unfolding dist_blinfun_def open_blinfun_def sgn_blinfun_def uniformity_blinfun_def
  apply (rule refl | (transfer, force simp: onorm_triangle onorm_scaleR onorm_eq_0 algebra_simps))+
  done

end

declare uniformity_Abort[where 'a="('a :: real_normed_vector) \<Rightarrow>\<^sub>L ('b :: real_normed_vector)", code]

lemma norm_blinfun_eqI:
  assumes "n \<le> norm (blinfun_apply f x) / norm x"
  assumes "\<And>x. norm (blinfun_apply f x) \<le> n * norm x"
  assumes "0 \<le> n"
  shows "norm f = n"
  by (auto simp: norm_blinfun_def
    intro!: antisym onorm_bound assms order_trans[OF _ le_onorm]
    bounded_linear_intros)

lemma norm_blinfun: "norm (blinfun_apply f x) \<le> norm f * norm x"
  by transfer (rule onorm)

lemma norm_blinfun_bound: "0 \<le> b \<Longrightarrow> (\<And>x. norm (blinfun_apply f x) \<le> b * norm x) \<Longrightarrow> norm f \<le> b"
  by transfer (rule onorm_bound)

lemma bounded_bilinear_blinfun_apply[bounded_bilinear]: "bounded_bilinear blinfun_apply"
proof
  fix f g::"'a \<Rightarrow>\<^sub>L 'b" and a b::'a and r::real
  show "(f + g) a = f a + g a" "(r *\<^sub>R f) a = r *\<^sub>R f a"
    by (transfer, simp)+
  interpret bounded_linear f for f::"'a \<Rightarrow>\<^sub>L 'b"
    by (auto intro!: bounded_linear_intros)
  show "f (a + b) = f a + f b" "f (r *\<^sub>R a) = r *\<^sub>R f a"
    by (simp_all add: add scaleR)
  show "\<exists>K. \<forall>a b. norm (blinfun_apply a b) \<le> norm a * norm b * K"
    by (auto intro!: exI[where x=1] norm_blinfun)
qed

interpretation blinfun: bounded_bilinear blinfun_apply
  by (rule bounded_bilinear_blinfun_apply)

lemmas bounded_linear_apply_blinfun[intro, simp] = blinfun.bounded_linear_left


context bounded_bilinear
begin

named_theorems bilinear_simps

lemmas [bilinear_simps] =
  add_left
  add_right
  diff_left
  diff_right
  minus_left
  minus_right
  scaleR_left
  scaleR_right
  zero_left
  zero_right
  sum_left
  sum_right

end


instance blinfun :: (banach, banach) banach
proof
  fix X::"nat \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  assume "Cauchy X"
  {
    fix x::'a
    {
      fix x::'a
      assume "norm x \<le> 1"
      have "Cauchy (\<lambda>n. X n x)"
      proof (rule CauchyI)
        fix e::real
        assume "0 < e"
        from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
          where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
          by auto
        show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m x - X n x) < e"
        proof (safe intro!: exI[where x=M])
          fix m n
          assume le: "M \<le> m" "M \<le> n"
          have "norm (X m x - X n x) = norm ((X m - X n) x)"
            by (simp add: blinfun.bilinear_simps)
          also have "\<dots> \<le> norm (X m - X n) * norm x"
             by (rule norm_blinfun)
          also have "\<dots> \<le> norm (X m - X n) * 1"
            using \<open>norm x \<le> 1\<close> norm_ge_zero by (rule mult_left_mono)
          also have "\<dots> = norm (X m - X n)" by simp
          also have "\<dots> < e" using le by fact
          finally show "norm (X m x - X n x) < e" .
        qed
      qed
      hence "convergent (\<lambda>n. X n x)"
        by (metis Cauchy_convergent_iff)
    } note convergent_norm1 = this
    define y where "y = x /\<^sub>R norm x"
    have y: "norm y \<le> 1" and xy: "x = norm x *\<^sub>R y"
      by (simp_all add: y_def inverse_eq_divide)
    have "convergent (\<lambda>n. norm x *\<^sub>R X n y)"
      by (intro bounded_bilinear.convergent[OF bounded_bilinear_scaleR] convergent_const
        convergent_norm1 y)
    also have "(\<lambda>n. norm x *\<^sub>R X n y) = (\<lambda>n. X n x)"
      by (subst xy) (simp add: blinfun.bilinear_simps)
    finally have "convergent (\<lambda>n. X n x)" .
  }
  then obtain v where v: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> v x"
    unfolding convergent_def
    by metis

  have "Cauchy (\<lambda>n. norm (X n))"
  proof (rule CauchyI)
    fix e::real
    assume "e > 0"
    from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
      where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
      by auto
    show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (norm (X m) - norm (X n)) < e"
    proof (safe intro!: exI[where x=M])
      fix m n assume mn: "m \<ge> M" "n \<ge> M"
      have "norm (norm (X m) - norm (X n)) \<le> norm (X m - X n)"
        by (metis norm_triangle_ineq3 real_norm_def)
      also have "\<dots> < e" using mn by fact
      finally show "norm (norm (X m) - norm (X n)) < e" .
    qed
  qed
  then obtain K where K: "(\<lambda>n. norm (X n)) \<longlonglongrightarrow> K"
    unfolding Cauchy_convergent_iff convergent_def
    by metis

  have "bounded_linear v"
  proof
    fix x y and r::real
    from tendsto_add[OF v[of x] v [of y]] v[of "x + y", unfolded blinfun.bilinear_simps]
      tendsto_scaleR[OF tendsto_const[of r] v[of x]] v[of "r *\<^sub>R x", unfolded blinfun.bilinear_simps]
    show "v (x + y) = v x + v y" "v (r *\<^sub>R x) = r *\<^sub>R v x"
      by (metis (poly_guards_query) LIMSEQ_unique)+
    show "\<exists>K. \<forall>x. norm (v x) \<le> norm x * K"
    proof (safe intro!: exI[where x=K])
      fix x
      have "norm (v x) \<le> K * norm x"
        by (rule tendsto_le[OF _ tendsto_mult[OF K tendsto_const] tendsto_norm[OF v]])
          (auto simp: norm_blinfun)
      thus "norm (v x) \<le> norm x * K"
        by (simp add: ac_simps)
    qed
  qed
  hence Bv: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> Blinfun v x"
    by (auto simp: bounded_linear_Blinfun_apply v)

  have "X \<longlonglongrightarrow> Blinfun v"
  proof (rule LIMSEQ_I)
    fix r::real assume "r > 0"
    define r' where "r' = r / 2"
    have "0 < r'" "r' < r" using \<open>r > 0\<close> by (simp_all add: r'_def)
    from CauchyD[OF \<open>Cauchy X\<close> \<open>r' > 0\<close>]
    obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < r'"
      by metis
    show "\<exists>no. \<forall>n\<ge>no. norm (X n - Blinfun v) < r"
    proof (safe intro!: exI[where x=M])
      fix n assume n: "M \<le> n"
      have "norm (X n - Blinfun v) \<le> r'"
      proof (rule norm_blinfun_bound)
        fix x
        have "eventually (\<lambda>m. m \<ge> M) sequentially"
          by (metis eventually_ge_at_top)
        hence ev_le: "eventually (\<lambda>m. norm (X n x - X m x) \<le> r' * norm x) sequentially"
        proof eventually_elim
          case (elim m)
          have "norm (X n x - X m x) = norm ((X n - X m) x)"
            by (simp add: blinfun.bilinear_simps)
          also have "\<dots> \<le> norm ((X n - X m)) * norm x"
            by (rule norm_blinfun)
          also have "\<dots> \<le> r' * norm x"
            using M[OF n elim] by (simp add: mult_right_mono)
          finally show ?case .
        qed
        have tendsto_v: "(\<lambda>m. norm (X n x - X m x)) \<longlonglongrightarrow> norm (X n x - Blinfun v x)"
          by (auto intro!: tendsto_intros Bv)
        show "norm ((X n - Blinfun v) x) \<le> r' * norm x"
          by (auto intro!: tendsto_upperbound tendsto_v ev_le simp: blinfun.bilinear_simps)
      qed (simp add: \<open>0 < r'\<close> less_imp_le)
      thus "norm (X n - Blinfun v) < r"
        by (metis \<open>r' < r\<close> le_less_trans)
    qed
  qed
  thus "convergent X"
    by (rule convergentI)
qed

subsection \<open>On Euclidean Space\<close>

lemma Zfun_sum:
  assumes "finite s"
  assumes f: "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F"
  shows "Zfun (\<lambda>x. sum (\<lambda>i. f i x) s) F"
  using assms by induct (auto intro!: Zfun_zero Zfun_add)

lemma norm_blinfun_euclidean_le:
  fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
  shows "norm a \<le> sum (\<lambda>x. norm (a x)) Basis"
  apply (rule norm_blinfun_bound)
   apply (simp add: sum_nonneg)
  apply (subst euclidean_representation[symmetric, where 'a='a])
  apply (simp only: blinfun.bilinear_simps sum_distrib_right)
  apply (rule order.trans[OF norm_sum sum_mono])
  apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
  done

lemma tendsto_componentwise1:
  fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
    and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  assumes "(\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j) \<longlongrightarrow> a j) F)"
  shows "(b \<longlongrightarrow> a) F"
proof -
  have "\<And>j. j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x j - a j)) F"
    using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
  hence "Zfun (\<lambda>x. \<Sum>j\<in>Basis. norm (b x j - a j)) F"
    by (auto intro!: Zfun_sum)
  thus ?thesis
    unfolding tendsto_Zfun_iff
    by (rule Zfun_le)
      (auto intro!: order_trans[OF norm_blinfun_euclidean_le] simp: blinfun.bilinear_simps)
qed

lift_definition
  blinfun_of_matrix::"('b::euclidean_space \<Rightarrow> 'a::euclidean_space \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  is "\<lambda>a x. \<Sum>i\<in>Basis. \<Sum>j\<in>Basis. ((x \<bullet> j) * a i j) *\<^sub>R i"
  by (intro bounded_linear_intros)

lemma blinfun_of_matrix_works:
  fixes f::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
  shows "blinfun_of_matrix (\<lambda>i j. (f j) \<bullet> i) = f"
proof (transfer, rule,  rule euclidean_eqI)
  fix f::"'a \<Rightarrow> 'b" and x::'a and b::'b assume "bounded_linear f" and b: "b \<in> Basis"
  then interpret bounded_linear f by simp
  have "(\<Sum>j\<in>Basis. \<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j)) *\<^sub>R j) \<bullet> b
    = (\<Sum>j\<in>Basis. if j = b then (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j))) else 0)"
    using b
    by (simp add: inner_sum_left inner_Basis if_distrib cong: if_cong) (simp add: sum.commute)
  also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> b)))"
    using b by (simp add: sum.delta)
  also have "\<dots> = f x \<bullet> b"
    by (metis (mono_tags, lifting) Linear_Algebra.linear_componentwise linear_axioms)
  finally show "(\<Sum>j\<in>Basis. \<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j)) *\<^sub>R j) \<bullet> b = f x \<bullet> b" .
qed

lemma blinfun_of_matrix_apply:
  "blinfun_of_matrix a x = (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. ((x \<bullet> j) * a i j) *\<^sub>R i)"
  by transfer simp

lemma blinfun_of_matrix_minus: "blinfun_of_matrix x - blinfun_of_matrix y = blinfun_of_matrix (x - y)"
  by transfer (auto simp: algebra_simps sum_subtractf)

lemma norm_blinfun_of_matrix:
  "norm (blinfun_of_matrix a) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>a i j\<bar>)"
  apply (rule norm_blinfun_bound)
   apply (simp add: sum_nonneg)
  apply (simp only: blinfun_of_matrix_apply sum_distrib_right)
  apply (rule order_trans[OF norm_sum sum_mono])
  apply (rule order_trans[OF norm_sum sum_mono])
  apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
  done

lemma tendsto_blinfun_of_matrix:
  assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
  shows "((\<lambda>n. blinfun_of_matrix (b n)) \<longlongrightarrow> blinfun_of_matrix a) F"
proof -
  have "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x i j - a i j)) F"
    using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
  hence "Zfun (\<lambda>x. (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>b x i j - a i j\<bar>)) F"
    by (auto intro!: Zfun_sum)
  thus ?thesis
    unfolding tendsto_Zfun_iff blinfun_of_matrix_minus
    by (rule Zfun_le) (auto intro!: order_trans[OF norm_blinfun_of_matrix])
qed

lemma tendsto_componentwise:
  fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
    and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  shows "(\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j \<bullet> i) \<longlongrightarrow> a j \<bullet> i) F) \<Longrightarrow> (b \<longlongrightarrow> a) F"
  apply (subst blinfun_of_matrix_works[of a, symmetric])
  apply (subst blinfun_of_matrix_works[of "b x" for x, symmetric, abs_def])
  by (rule tendsto_blinfun_of_matrix)

lemma
  continuous_blinfun_componentwiseI:
  fixes f:: "'b::t2_space \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'c::euclidean_space"
  assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> continuous F (\<lambda>x. (f x) j \<bullet> i)"
  shows "continuous F f"
  using assms by (auto simp: continuous_def intro!: tendsto_componentwise)

lemma
  continuous_blinfun_componentwiseI1:
  fixes f:: "'b::t2_space \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'c::real_normed_vector"
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous F (\<lambda>x. f x i)"
  shows "continuous F f"
  using assms by (auto simp: continuous_def intro!: tendsto_componentwise1)

lemma bounded_linear_blinfun_matrix: "bounded_linear (\<lambda>x. (x::_\<Rightarrow>\<^sub>L _) j \<bullet> i)"
  by (auto intro!: bounded_linearI' bounded_linear_intros)

lemma continuous_blinfun_matrix:
  fixes f:: "'b::t2_space \<Rightarrow> 'a::real_normed_vector \<Rightarrow>\<^sub>L 'c::real_inner"
  assumes "continuous F f"
  shows "continuous F (\<lambda>x. (f x) j \<bullet> i)"
  by (rule bounded_linear.continuous[OF bounded_linear_blinfun_matrix assms])

lemma continuous_on_blinfun_matrix:
  fixes f::"'a::t2_space \<Rightarrow> 'b::real_normed_vector \<Rightarrow>\<^sub>L 'c::real_inner"
  assumes "continuous_on S f"
  shows "continuous_on S (\<lambda>x. (f x) j \<bullet> i)"
  using assms
  by (auto simp: continuous_on_eq_continuous_within continuous_blinfun_matrix)

lemma continuous_on_blinfun_of_matrix[continuous_intros]:
  assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> continuous_on S (\<lambda>s. g s i j)"
  shows "continuous_on S (\<lambda>s. blinfun_of_matrix (g s))"
  using assms
  by (auto simp: continuous_on intro!: tendsto_blinfun_of_matrix)

lemma mult_if_delta:
  "(if P then (1::'a::comm_semiring_1) else 0) * q = (if P then q else 0)"
  by auto

lemma compact_blinfun_lemma:
  fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
  assumes "bounded (range f)"
  shows "\<forall>d\<subseteq>Basis. \<exists>l::'a \<Rightarrow>\<^sub>L 'b. \<exists> r.
    subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
  by (rule compact_lemma_general[where unproj = "\<lambda>e. blinfun_of_matrix (\<lambda>i j. e j \<bullet> i)"])
   (auto intro!: euclidean_eqI[where 'a='b] bounded_linear_image assms
    simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta sum.delta'
      scaleR_sum_left[symmetric])

lemma blinfun_euclidean_eqI: "(\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
  apply (auto intro!: blinfun_eqI)
  apply (subst (2) euclidean_representation[symmetric, where 'a='a])
  apply (subst (1) euclidean_representation[symmetric, where 'a='a])
  apply (simp add: blinfun.bilinear_simps)
  done

lemma Blinfun_eq_matrix: "bounded_linear f \<Longrightarrow> Blinfun f = blinfun_of_matrix (\<lambda>i j. f j \<bullet> i)"
  by (intro blinfun_euclidean_eqI)
     (auto simp: blinfun_of_matrix_apply bounded_linear_Blinfun_apply inner_Basis if_distrib
      cond_application_beta sum.delta' euclidean_representation
      cong: if_cong)

text \<open>TODO: generalize (via @{thm compact_cball})?\<close>
instance blinfun :: (euclidean_space, euclidean_space) heine_borel
proof
  fix f :: "nat \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
  assume f: "bounded (range f)"
  then obtain l::"'a \<Rightarrow>\<^sub>L 'b" and r where r: "subseq r"
    and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) i) (l i) < e) sequentially"
    using compact_blinfun_lemma [OF f] by blast
  {
    fix e::real
    let ?d = "real_of_nat DIM('a) * real_of_nat DIM('b)"
    assume "e > 0"
    hence "e / ?d > 0" by (simp add: DIM_positive)
    with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) i) (l i) < e / ?d) sequentially"
      by simp
    moreover
    {
      fix n
      assume n: "\<forall>i\<in>Basis. dist (f (r n) i) (l i) < e / ?d"
      have "norm (f (r n) - l) = norm (blinfun_of_matrix (\<lambda>i j. (f (r n) - l) j \<bullet> i))"
        unfolding blinfun_of_matrix_works ..
      also note norm_blinfun_of_matrix
      also have "(\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) <
        (\<Sum>i\<in>(Basis::'b set). e / real_of_nat DIM('b))"
      proof (rule sum_strict_mono)
        fix i::'b assume i: "i \<in> Basis"
        have "(\<Sum>j::'a\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) < (\<Sum>j::'a\<in>Basis. e / ?d)"
        proof (rule sum_strict_mono)
          fix j::'a assume j: "j \<in> Basis"
          have "\<bar>(f (r n) - l) j \<bullet> i\<bar> \<le> norm ((f (r n) - l) j)"
            by (simp add: Basis_le_norm i)
          also have "\<dots> < e / ?d"
            using n i j by (auto simp: dist_norm blinfun.bilinear_simps)
          finally show "\<bar>(f (r n) - l) j \<bullet> i\<bar> < e / ?d" by simp
        qed simp_all
        also have "\<dots> \<le> e / real_of_nat DIM('b)"
          by simp
        finally show "(\<Sum>j\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) < e / real_of_nat DIM('b)"
          by simp
      qed simp_all
      also have "\<dots> \<le> e" by simp
      finally have "dist (f (r n)) l < e"
        by (auto simp: dist_norm)
    }
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
      using eventually_elim2 by force
  }
  then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
    unfolding o_def tendsto_iff by simp
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
    by auto
qed


subsection \<open>concrete bounded linear functions\<close>

lemma transfer_bounded_bilinear_bounded_linearI:
  assumes "g = (\<lambda>i x. (blinfun_apply (f i) x))"
  shows "bounded_bilinear g = bounded_linear f"
proof
  assume "bounded_bilinear g"
  then interpret bounded_bilinear f by (simp add: assms)
  show "bounded_linear f"
  proof (unfold_locales, safe intro!: blinfun_eqI)
    fix i
    show "f (x + y) i = (f x + f y) i" "f (r *\<^sub>R x) i = (r *\<^sub>R f x) i" for r x y
      by (auto intro!: blinfun_eqI simp: blinfun.bilinear_simps)
    from _ nonneg_bounded show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
      by (rule ex_reg) (auto intro!: onorm_bound simp: norm_blinfun.rep_eq ac_simps)
  qed
qed (auto simp: assms intro!: blinfun.comp)

lemma transfer_bounded_bilinear_bounded_linear[transfer_rule]:
  "(rel_fun (rel_fun op = (pcr_blinfun op = op =)) op =) bounded_bilinear bounded_linear"
  by (auto simp: pcr_blinfun_def cr_blinfun_def rel_fun_def OO_def
    intro!: transfer_bounded_bilinear_bounded_linearI)

context bounded_bilinear
begin

lift_definition prod_left::"'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'c" is "(\<lambda>b a. prod a b)"
  by (rule bounded_linear_left)
declare prod_left.rep_eq[simp]

lemma bounded_linear_prod_left[bounded_linear]: "bounded_linear prod_left"
  by transfer (rule flip)

lift_definition prod_right::"'a \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c" is "(\<lambda>a b. prod a b)"
  by (rule bounded_linear_right)
declare prod_right.rep_eq[simp]

lemma bounded_linear_prod_right[bounded_linear]: "bounded_linear prod_right"
  by transfer (rule bounded_bilinear_axioms)

end

lift_definition id_blinfun::"'a::real_normed_vector \<Rightarrow>\<^sub>L 'a" is "\<lambda>x. x"
  by (rule bounded_linear_ident)

lemmas blinfun_apply_id_blinfun[simp] = id_blinfun.rep_eq

lemma norm_blinfun_id[simp]:
  "norm (id_blinfun::'a::{real_normed_vector, perfect_space} \<Rightarrow>\<^sub>L 'a) = 1"
  by transfer (auto simp: onorm_id)

lemma norm_blinfun_id_le:
  "norm (id_blinfun::'a::real_normed_vector \<Rightarrow>\<^sub>L 'a) \<le> 1"
  by transfer (auto simp: onorm_id_le)


lift_definition fst_blinfun::"('a::real_normed_vector \<times> 'b::real_normed_vector) \<Rightarrow>\<^sub>L 'a" is fst
  by (rule bounded_linear_fst)

lemma blinfun_apply_fst_blinfun[simp]: "blinfun_apply fst_blinfun = fst"
  by transfer (rule refl)


lift_definition snd_blinfun::"('a::real_normed_vector \<times> 'b::real_normed_vector) \<Rightarrow>\<^sub>L 'b" is snd
  by (rule bounded_linear_snd)

lemma blinfun_apply_snd_blinfun[simp]: "blinfun_apply snd_blinfun = snd"
  by transfer (rule refl)


lift_definition blinfun_compose::
  "'a::real_normed_vector \<Rightarrow>\<^sub>L 'b::real_normed_vector \<Rightarrow>
    'c::real_normed_vector \<Rightarrow>\<^sub>L 'a \<Rightarrow>
    'c \<Rightarrow>\<^sub>L 'b" (infixl "o\<^sub>L" 55) is "op o"
  parametric comp_transfer
  unfolding o_def
  by (rule bounded_linear_compose)

lemma blinfun_apply_blinfun_compose[simp]: "(a o\<^sub>L b) c = a (b c)"
  by (simp add: blinfun_compose.rep_eq)

lemma norm_blinfun_compose:
  "norm (f o\<^sub>L g) \<le> norm f * norm g"
  by transfer (rule onorm_compose)

lemma bounded_bilinear_blinfun_compose[bounded_bilinear]: "bounded_bilinear op o\<^sub>L"
  by unfold_locales
    (auto intro!: blinfun_eqI exI[where x=1] simp: blinfun.bilinear_simps norm_blinfun_compose)

lemma blinfun_compose_zero[simp]:
  "blinfun_compose 0 = (\<lambda>_. 0)"
  "blinfun_compose x 0 = 0"
  by (auto simp: blinfun.bilinear_simps intro!: blinfun_eqI)


lift_definition blinfun_inner_right::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "op \<bullet>"
  by (rule bounded_linear_inner_right)
declare blinfun_inner_right.rep_eq[simp]

lemma bounded_linear_blinfun_inner_right[bounded_linear]: "bounded_linear blinfun_inner_right"
  by transfer (rule bounded_bilinear_inner)


lift_definition blinfun_inner_left::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "\<lambda>x y. y \<bullet> x"
  by (rule bounded_linear_inner_left)
declare blinfun_inner_left.rep_eq[simp]

lemma bounded_linear_blinfun_inner_left[bounded_linear]: "bounded_linear blinfun_inner_left"
  by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_inner])


lift_definition blinfun_scaleR_right::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_vector" is "op *\<^sub>R"
  by (rule bounded_linear_scaleR_right)
declare blinfun_scaleR_right.rep_eq[simp]

lemma bounded_linear_blinfun_scaleR_right[bounded_linear]: "bounded_linear blinfun_scaleR_right"
  by transfer (rule bounded_bilinear_scaleR)


lift_definition blinfun_scaleR_left::"'a::real_normed_vector \<Rightarrow> real \<Rightarrow>\<^sub>L 'a" is "\<lambda>x y. y *\<^sub>R x"
  by (rule bounded_linear_scaleR_left)
lemmas [simp] = blinfun_scaleR_left.rep_eq

lemma bounded_linear_blinfun_scaleR_left[bounded_linear]: "bounded_linear blinfun_scaleR_left"
  by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_scaleR])


lift_definition blinfun_mult_right::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_algebra" is "op *"
  by (rule bounded_linear_mult_right)
declare blinfun_mult_right.rep_eq[simp]

lemma bounded_linear_blinfun_mult_right[bounded_linear]: "bounded_linear blinfun_mult_right"
  by transfer (rule bounded_bilinear_mult)


lift_definition blinfun_mult_left::"'a::real_normed_algebra \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a" is "\<lambda>x y. y * x"
  by (rule bounded_linear_mult_left)
lemmas [simp] = blinfun_mult_left.rep_eq

lemma bounded_linear_blinfun_mult_left[bounded_linear]: "bounded_linear blinfun_mult_left"
  by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_mult])

end