src/HOL/Analysis/Bounded_Linear_Function.thy
 author nipkow Sat Dec 29 15:43:53 2018 +0100 (6 months ago) changeset 69529 4ab9657b3257 parent 69260 0a9688695a1b child 69597 ff784d5a5bfb permissions -rw-r--r--
capitalize proper names in lemma names
```     1 (*  Title:      HOL/Analysis/Bounded_Linear_Function.thy
```
```     2     Author:     Fabian Immler, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Bounded Linear Function\<close>
```
```     6
```
```     7 theory Bounded_Linear_Function
```
```     8 imports
```
```     9   Topology_Euclidean_Space
```
```    10   Operator_Norm
```
```    11   Uniform_Limit
```
```    12 begin
```
```    13
```
```    14 lemma onorm_componentwise:
```
```    15   assumes "bounded_linear f"
```
```    16   shows "onorm f \<le> (\<Sum>i\<in>Basis. norm (f i))"
```
```    17 proof -
```
```    18   {
```
```    19     fix i::'a
```
```    20     assume "i \<in> Basis"
```
```    21     hence "onorm (\<lambda>x. (x \<bullet> i) *\<^sub>R f i) \<le> onorm (\<lambda>x. (x \<bullet> i)) * norm (f i)"
```
```    22       by (auto intro!: onorm_scaleR_left_lemma bounded_linear_inner_left)
```
```    23     also have "\<dots> \<le>  norm i * norm (f i)"
```
```    24       by (rule mult_right_mono)
```
```    25         (auto simp: ac_simps Cauchy_Schwarz_ineq2 intro!: onorm_le)
```
```    26     finally have "onorm (\<lambda>x. (x \<bullet> i) *\<^sub>R f i) \<le> norm (f i)" using \<open>i \<in> Basis\<close>
```
```    27       by simp
```
```    28   } hence "onorm (\<lambda>x. \<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<le> (\<Sum>i\<in>Basis. norm (f i))"
```
```    29     by (auto intro!: order_trans[OF onorm_sum_le] bounded_linear_scaleR_const
```
```    30       sum_mono bounded_linear_inner_left)
```
```    31   also have "(\<lambda>x. \<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) = (\<lambda>x. f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i))"
```
```    32     by (simp add: linear_sum bounded_linear.linear assms linear_simps)
```
```    33   also have "\<dots> = f"
```
```    34     by (simp add: euclidean_representation)
```
```    35   finally show ?thesis .
```
```    36 qed
```
```    37
```
```    38 lemmas onorm_componentwise_le = order_trans[OF onorm_componentwise]
```
```    39
```
```    40 subsection%unimportant \<open>Intro rules for @{term bounded_linear}\<close>
```
```    41
```
```    42 named_theorems bounded_linear_intros
```
```    43
```
```    44 lemma onorm_inner_left:
```
```    45   assumes "bounded_linear r"
```
```    46   shows "onorm (\<lambda>x. r x \<bullet> f) \<le> onorm r * norm f"
```
```    47 proof (rule onorm_bound)
```
```    48   fix x
```
```    49   have "norm (r x \<bullet> f) \<le> norm (r x) * norm f"
```
```    50     by (simp add: Cauchy_Schwarz_ineq2)
```
```    51   also have "\<dots> \<le> onorm r * norm x * norm f"
```
```    52     by (intro mult_right_mono onorm assms norm_ge_zero)
```
```    53   finally show "norm (r x \<bullet> f) \<le> onorm r * norm f * norm x"
```
```    54     by (simp add: ac_simps)
```
```    55 qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms)
```
```    56
```
```    57 lemma onorm_inner_right:
```
```    58   assumes "bounded_linear r"
```
```    59   shows "onorm (\<lambda>x. f \<bullet> r x) \<le> norm f * onorm r"
```
```    60   apply (subst inner_commute)
```
```    61   apply (rule onorm_inner_left[OF assms, THEN order_trans])
```
```    62   apply simp
```
```    63   done
```
```    64
```
```    65 lemmas [bounded_linear_intros] =
```
```    66   bounded_linear_zero
```
```    67   bounded_linear_add
```
```    68   bounded_linear_const_mult
```
```    69   bounded_linear_mult_const
```
```    70   bounded_linear_scaleR_const
```
```    71   bounded_linear_const_scaleR
```
```    72   bounded_linear_ident
```
```    73   bounded_linear_sum
```
```    74   bounded_linear_Pair
```
```    75   bounded_linear_sub
```
```    76   bounded_linear_fst_comp
```
```    77   bounded_linear_snd_comp
```
```    78   bounded_linear_inner_left_comp
```
```    79   bounded_linear_inner_right_comp
```
```    80
```
```    81
```
```    82 subsection%unimportant \<open>declaration of derivative/continuous/tendsto introduction rules for bounded linear functions\<close>
```
```    83
```
```    84 attribute_setup bounded_linear =
```
```    85   \<open>Scan.succeed (Thm.declaration_attribute (fn thm =>
```
```    86     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
```
```    87       [
```
```    88         (@{thm bounded_linear.has_derivative}, @{named_theorems derivative_intros}),
```
```    89         (@{thm bounded_linear.tendsto}, @{named_theorems tendsto_intros}),
```
```    90         (@{thm bounded_linear.continuous}, @{named_theorems continuous_intros}),
```
```    91         (@{thm bounded_linear.continuous_on}, @{named_theorems continuous_intros}),
```
```    92         (@{thm bounded_linear.uniformly_continuous_on}, @{named_theorems continuous_intros}),
```
```    93         (@{thm bounded_linear_compose}, @{named_theorems bounded_linear_intros})
```
```    94       ]))\<close>
```
```    95
```
```    96 attribute_setup bounded_bilinear =
```
```    97   \<open>Scan.succeed (Thm.declaration_attribute (fn thm =>
```
```    98     fold (fn (r, s) => Named_Theorems.add_thm s (thm RS r))
```
```    99       [
```
```   100         (@{thm bounded_bilinear.FDERIV}, @{named_theorems derivative_intros}),
```
```   101         (@{thm bounded_bilinear.tendsto}, @{named_theorems tendsto_intros}),
```
```   102         (@{thm bounded_bilinear.continuous}, @{named_theorems continuous_intros}),
```
```   103         (@{thm bounded_bilinear.continuous_on}, @{named_theorems continuous_intros}),
```
```   104         (@{thm bounded_linear_compose[OF bounded_bilinear.bounded_linear_left]},
```
```   105           @{named_theorems bounded_linear_intros}),
```
```   106         (@{thm bounded_linear_compose[OF bounded_bilinear.bounded_linear_right]},
```
```   107           @{named_theorems bounded_linear_intros}),
```
```   108         (@{thm bounded_linear.uniformly_continuous_on[OF bounded_bilinear.bounded_linear_left]},
```
```   109           @{named_theorems continuous_intros}),
```
```   110         (@{thm bounded_linear.uniformly_continuous_on[OF bounded_bilinear.bounded_linear_right]},
```
```   111           @{named_theorems continuous_intros})
```
```   112       ]))\<close>
```
```   113
```
```   114
```
```   115 subsection \<open>Type of bounded linear functions\<close>
```
```   116
```
```   117 typedef%important (overloaded) ('a, 'b) blinfun ("(_ \<Rightarrow>\<^sub>L /_)" [22, 21] 21) =
```
```   118   "{f::'a::real_normed_vector\<Rightarrow>'b::real_normed_vector. bounded_linear f}"
```
```   119   morphisms blinfun_apply Blinfun
```
```   120   by (blast intro: bounded_linear_intros)
```
```   121
```
```   122 declare [[coercion
```
```   123     "blinfun_apply :: ('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'b"]]
```
```   124
```
```   125 lemma bounded_linear_blinfun_apply[bounded_linear_intros]:
```
```   126   "bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. blinfun_apply f (g x))"
```
```   127   by (metis blinfun_apply mem_Collect_eq bounded_linear_compose)
```
```   128
```
```   129 setup_lifting type_definition_blinfun
```
```   130
```
```   131 lemma blinfun_eqI: "(\<And>i. blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
```
```   132   by transfer auto
```
```   133
```
```   134 lemma bounded_linear_Blinfun_apply: "bounded_linear f \<Longrightarrow> blinfun_apply (Blinfun f) = f"
```
```   135   by (auto simp: Blinfun_inverse)
```
```   136
```
```   137
```
```   138 subsection \<open>Type class instantiations\<close>
```
```   139
```
```   140 instantiation blinfun :: (real_normed_vector, real_normed_vector) real_normed_vector
```
```   141 begin
```
```   142
```
```   143 lift_definition%important norm_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> real" is onorm .
```
```   144
```
```   145 lift_definition minus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   146   is "\<lambda>f g x. f x - g x"
```
```   147   by (rule bounded_linear_sub)
```
```   148
```
```   149 definition dist_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> real"
```
```   150   where "dist_blinfun a b = norm (a - b)"
```
```   151
```
```   152 definition [code del]:
```
```   153   "(uniformity :: (('a \<Rightarrow>\<^sub>L 'b) \<times> ('a \<Rightarrow>\<^sub>L 'b)) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
```
```   154
```
```   155 definition open_blinfun :: "('a \<Rightarrow>\<^sub>L 'b) set \<Rightarrow> bool"
```
```   156   where [code del]: "open_blinfun S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
```
```   157
```
```   158 lift_definition uminus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" is "\<lambda>f x. - f x"
```
```   159   by (rule bounded_linear_minus)
```
```   160
```
```   161 lift_definition%important zero_blinfun :: "'a \<Rightarrow>\<^sub>L 'b" is "\<lambda>x. 0"
```
```   162   by (rule bounded_linear_zero)
```
```   163
```
```   164 lift_definition%important plus_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   165   is "\<lambda>f g x. f x + g x"
```
```   166   by (metis bounded_linear_add)
```
```   167
```
```   168 lift_definition%important 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"
```
```   169   by (metis bounded_linear_compose bounded_linear_scaleR_right)
```
```   170
```
```   171 definition sgn_blinfun :: "'a \<Rightarrow>\<^sub>L 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   172   where "sgn_blinfun x = scaleR (inverse (norm x)) x"
```
```   173
```
```   174 instance
```
```   175   apply standard
```
```   176   unfolding dist_blinfun_def open_blinfun_def sgn_blinfun_def uniformity_blinfun_def
```
```   177   apply (rule refl | (transfer, force simp: onorm_triangle onorm_scaleR onorm_eq_0 algebra_simps))+
```
```   178   done
```
```   179
```
```   180 end
```
```   181
```
```   182 declare uniformity_Abort[where 'a="('a :: real_normed_vector) \<Rightarrow>\<^sub>L ('b :: real_normed_vector)", code]
```
```   183
```
```   184 lemma norm_blinfun_eqI:
```
```   185   assumes "n \<le> norm (blinfun_apply f x) / norm x"
```
```   186   assumes "\<And>x. norm (blinfun_apply f x) \<le> n * norm x"
```
```   187   assumes "0 \<le> n"
```
```   188   shows "norm f = n"
```
```   189   by (auto simp: norm_blinfun_def
```
```   190     intro!: antisym onorm_bound assms order_trans[OF _ le_onorm]
```
```   191     bounded_linear_intros)
```
```   192
```
```   193 lemma norm_blinfun: "norm (blinfun_apply f x) \<le> norm f * norm x"
```
```   194   by transfer (rule onorm)
```
```   195
```
```   196 lemma norm_blinfun_bound: "0 \<le> b \<Longrightarrow> (\<And>x. norm (blinfun_apply f x) \<le> b * norm x) \<Longrightarrow> norm f \<le> b"
```
```   197   by transfer (rule onorm_bound)
```
```   198
```
```   199 lemma bounded_bilinear_blinfun_apply[bounded_bilinear]: "bounded_bilinear blinfun_apply"
```
```   200 proof
```
```   201   fix f g::"'a \<Rightarrow>\<^sub>L 'b" and a b::'a and r::real
```
```   202   show "(f + g) a = f a + g a" "(r *\<^sub>R f) a = r *\<^sub>R f a"
```
```   203     by (transfer, simp)+
```
```   204   interpret bounded_linear f for f::"'a \<Rightarrow>\<^sub>L 'b"
```
```   205     by (auto intro!: bounded_linear_intros)
```
```   206   show "f (a + b) = f a + f b" "f (r *\<^sub>R a) = r *\<^sub>R f a"
```
```   207     by (simp_all add: add scaleR)
```
```   208   show "\<exists>K. \<forall>a b. norm (blinfun_apply a b) \<le> norm a * norm b * K"
```
```   209     by (auto intro!: exI[where x=1] norm_blinfun)
```
```   210 qed
```
```   211
```
```   212 interpretation blinfun: bounded_bilinear blinfun_apply
```
```   213   by (rule bounded_bilinear_blinfun_apply)
```
```   214
```
```   215 lemmas bounded_linear_apply_blinfun[intro, simp] = blinfun.bounded_linear_left
```
```   216
```
```   217
```
```   218 context bounded_bilinear
```
```   219 begin
```
```   220
```
```   221 named_theorems bilinear_simps
```
```   222
```
```   223 lemmas [bilinear_simps] =
```
```   224   add_left
```
```   225   add_right
```
```   226   diff_left
```
```   227   diff_right
```
```   228   minus_left
```
```   229   minus_right
```
```   230   scaleR_left
```
```   231   scaleR_right
```
```   232   zero_left
```
```   233   zero_right
```
```   234   sum_left
```
```   235   sum_right
```
```   236
```
```   237 end
```
```   238
```
```   239
```
```   240 instance blinfun :: (real_normed_vector, banach) banach
```
```   241 proof
```
```   242   fix X::"nat \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   243   assume "Cauchy X"
```
```   244   {
```
```   245     fix x::'a
```
```   246     {
```
```   247       fix x::'a
```
```   248       assume "norm x \<le> 1"
```
```   249       have "Cauchy (\<lambda>n. X n x)"
```
```   250       proof (rule CauchyI)
```
```   251         fix e::real
```
```   252         assume "0 < e"
```
```   253         from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
```
```   254           where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
```
```   255           by auto
```
```   256         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m x - X n x) < e"
```
```   257         proof (safe intro!: exI[where x=M])
```
```   258           fix m n
```
```   259           assume le: "M \<le> m" "M \<le> n"
```
```   260           have "norm (X m x - X n x) = norm ((X m - X n) x)"
```
```   261             by (simp add: blinfun.bilinear_simps)
```
```   262           also have "\<dots> \<le> norm (X m - X n) * norm x"
```
```   263              by (rule norm_blinfun)
```
```   264           also have "\<dots> \<le> norm (X m - X n) * 1"
```
```   265             using \<open>norm x \<le> 1\<close> norm_ge_zero by (rule mult_left_mono)
```
```   266           also have "\<dots> = norm (X m - X n)" by simp
```
```   267           also have "\<dots> < e" using le by fact
```
```   268           finally show "norm (X m x - X n x) < e" .
```
```   269         qed
```
```   270       qed
```
```   271       hence "convergent (\<lambda>n. X n x)"
```
```   272         by (metis Cauchy_convergent_iff)
```
```   273     } note convergent_norm1 = this
```
```   274     define y where "y = x /\<^sub>R norm x"
```
```   275     have y: "norm y \<le> 1" and xy: "x = norm x *\<^sub>R y"
```
```   276       by (simp_all add: y_def inverse_eq_divide)
```
```   277     have "convergent (\<lambda>n. norm x *\<^sub>R X n y)"
```
```   278       by (intro bounded_bilinear.convergent[OF bounded_bilinear_scaleR] convergent_const
```
```   279         convergent_norm1 y)
```
```   280     also have "(\<lambda>n. norm x *\<^sub>R X n y) = (\<lambda>n. X n x)"
```
```   281       by (subst xy) (simp add: blinfun.bilinear_simps)
```
```   282     finally have "convergent (\<lambda>n. X n x)" .
```
```   283   }
```
```   284   then obtain v where v: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> v x"
```
```   285     unfolding convergent_def
```
```   286     by metis
```
```   287
```
```   288   have "Cauchy (\<lambda>n. norm (X n))"
```
```   289   proof (rule CauchyI)
```
```   290     fix e::real
```
```   291     assume "e > 0"
```
```   292     from CauchyD[OF \<open>Cauchy X\<close> \<open>0 < e\<close>] obtain M
```
```   293       where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < e"
```
```   294       by auto
```
```   295     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (norm (X m) - norm (X n)) < e"
```
```   296     proof (safe intro!: exI[where x=M])
```
```   297       fix m n assume mn: "m \<ge> M" "n \<ge> M"
```
```   298       have "norm (norm (X m) - norm (X n)) \<le> norm (X m - X n)"
```
```   299         by (metis norm_triangle_ineq3 real_norm_def)
```
```   300       also have "\<dots> < e" using mn by fact
```
```   301       finally show "norm (norm (X m) - norm (X n)) < e" .
```
```   302     qed
```
```   303   qed
```
```   304   then obtain K where K: "(\<lambda>n. norm (X n)) \<longlonglongrightarrow> K"
```
```   305     unfolding Cauchy_convergent_iff convergent_def
```
```   306     by metis
```
```   307
```
```   308   have "bounded_linear v"
```
```   309   proof
```
```   310     fix x y and r::real
```
```   311     from tendsto_add[OF v[of x] v [of y]] v[of "x + y", unfolded blinfun.bilinear_simps]
```
```   312       tendsto_scaleR[OF tendsto_const[of r] v[of x]] v[of "r *\<^sub>R x", unfolded blinfun.bilinear_simps]
```
```   313     show "v (x + y) = v x + v y" "v (r *\<^sub>R x) = r *\<^sub>R v x"
```
```   314       by (metis (poly_guards_query) LIMSEQ_unique)+
```
```   315     show "\<exists>K. \<forall>x. norm (v x) \<le> norm x * K"
```
```   316     proof (safe intro!: exI[where x=K])
```
```   317       fix x
```
```   318       have "norm (v x) \<le> K * norm x"
```
```   319         by (rule tendsto_le[OF _ tendsto_mult[OF K tendsto_const] tendsto_norm[OF v]])
```
```   320           (auto simp: norm_blinfun)
```
```   321       thus "norm (v x) \<le> norm x * K"
```
```   322         by (simp add: ac_simps)
```
```   323     qed
```
```   324   qed
```
```   325   hence Bv: "\<And>x. (\<lambda>n. X n x) \<longlonglongrightarrow> Blinfun v x"
```
```   326     by (auto simp: bounded_linear_Blinfun_apply v)
```
```   327
```
```   328   have "X \<longlonglongrightarrow> Blinfun v"
```
```   329   proof (rule LIMSEQ_I)
```
```   330     fix r::real assume "r > 0"
```
```   331     define r' where "r' = r / 2"
```
```   332     have "0 < r'" "r' < r" using \<open>r > 0\<close> by (simp_all add: r'_def)
```
```   333     from CauchyD[OF \<open>Cauchy X\<close> \<open>r' > 0\<close>]
```
```   334     obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (X m - X n) < r'"
```
```   335       by metis
```
```   336     show "\<exists>no. \<forall>n\<ge>no. norm (X n - Blinfun v) < r"
```
```   337     proof (safe intro!: exI[where x=M])
```
```   338       fix n assume n: "M \<le> n"
```
```   339       have "norm (X n - Blinfun v) \<le> r'"
```
```   340       proof (rule norm_blinfun_bound)
```
```   341         fix x
```
```   342         have "eventually (\<lambda>m. m \<ge> M) sequentially"
```
```   343           by (metis eventually_ge_at_top)
```
```   344         hence ev_le: "eventually (\<lambda>m. norm (X n x - X m x) \<le> r' * norm x) sequentially"
```
```   345         proof eventually_elim
```
```   346           case (elim m)
```
```   347           have "norm (X n x - X m x) = norm ((X n - X m) x)"
```
```   348             by (simp add: blinfun.bilinear_simps)
```
```   349           also have "\<dots> \<le> norm ((X n - X m)) * norm x"
```
```   350             by (rule norm_blinfun)
```
```   351           also have "\<dots> \<le> r' * norm x"
```
```   352             using M[OF n elim] by (simp add: mult_right_mono)
```
```   353           finally show ?case .
```
```   354         qed
```
```   355         have tendsto_v: "(\<lambda>m. norm (X n x - X m x)) \<longlonglongrightarrow> norm (X n x - Blinfun v x)"
```
```   356           by (auto intro!: tendsto_intros Bv)
```
```   357         show "norm ((X n - Blinfun v) x) \<le> r' * norm x"
```
```   358           by (auto intro!: tendsto_upperbound tendsto_v ev_le simp: blinfun.bilinear_simps)
```
```   359       qed (simp add: \<open>0 < r'\<close> less_imp_le)
```
```   360       thus "norm (X n - Blinfun v) < r"
```
```   361         by (metis \<open>r' < r\<close> le_less_trans)
```
```   362     qed
```
```   363   qed
```
```   364   thus "convergent X"
```
```   365     by (rule convergentI)
```
```   366 qed
```
```   367
```
```   368 subsection%unimportant \<open>On Euclidean Space\<close>
```
```   369
```
```   370 lemma Zfun_sum:
```
```   371   assumes "finite s"
```
```   372   assumes f: "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F"
```
```   373   shows "Zfun (\<lambda>x. sum (\<lambda>i. f i x) s) F"
```
```   374   using assms by induct (auto intro!: Zfun_zero Zfun_add)
```
```   375
```
```   376 lemma norm_blinfun_euclidean_le:
```
```   377   fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
```
```   378   shows "norm a \<le> sum (\<lambda>x. norm (a x)) Basis"
```
```   379   apply (rule norm_blinfun_bound)
```
```   380    apply (simp add: sum_nonneg)
```
```   381   apply (subst euclidean_representation[symmetric, where 'a='a])
```
```   382   apply (simp only: blinfun.bilinear_simps sum_distrib_right)
```
```   383   apply (rule order.trans[OF norm_sum sum_mono])
```
```   384   apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
```
```   385   done
```
```   386
```
```   387 lemma tendsto_componentwise1:
```
```   388   fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::real_normed_vector"
```
```   389     and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   390   assumes "(\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n j) \<longlongrightarrow> a j) F)"
```
```   391   shows "(b \<longlongrightarrow> a) F"
```
```   392 proof -
```
```   393   have "\<And>j. j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x j - a j)) F"
```
```   394     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
```
```   395   hence "Zfun (\<lambda>x. \<Sum>j\<in>Basis. norm (b x j - a j)) F"
```
```   396     by (auto intro!: Zfun_sum)
```
```   397   thus ?thesis
```
```   398     unfolding tendsto_Zfun_iff
```
```   399     by (rule Zfun_le)
```
```   400       (auto intro!: order_trans[OF norm_blinfun_euclidean_le] simp: blinfun.bilinear_simps)
```
```   401 qed
```
```   402
```
```   403 lift_definition
```
```   404   blinfun_of_matrix::"('b::euclidean_space \<Rightarrow> 'a::euclidean_space \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   405   is "\<lambda>a x. \<Sum>i\<in>Basis. \<Sum>j\<in>Basis. ((x \<bullet> j) * a i j) *\<^sub>R i"
```
```   406   by (intro bounded_linear_intros)
```
```   407
```
```   408 lemma blinfun_of_matrix_works:
```
```   409   fixes f::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
```
```   410   shows "blinfun_of_matrix (\<lambda>i j. (f j) \<bullet> i) = f"
```
```   411 proof (transfer, rule,  rule euclidean_eqI)
```
```   412   fix f::"'a \<Rightarrow> 'b" and x::'a and b::'b assume "bounded_linear f" and b: "b \<in> Basis"
```
```   413   then interpret bounded_linear f by simp
```
```   414   have "(\<Sum>j\<in>Basis. \<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j)) *\<^sub>R j) \<bullet> b
```
```   415     = (\<Sum>j\<in>Basis. if j = b then (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> j))) else 0)"
```
```   416     using b
```
```   417     by (simp add: inner_sum_left inner_Basis if_distrib cong: if_cong) (simp add: sum.swap)
```
```   418   also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i * (f i \<bullet> b)))"
```
```   419     using b by (simp add: sum.delta)
```
```   420   also have "\<dots> = f x \<bullet> b"
```
```   421     by (metis (mono_tags, lifting) Linear_Algebra.linear_componentwise linear_axioms)
```
```   422   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" .
```
```   423 qed
```
```   424
```
```   425 lemma blinfun_of_matrix_apply:
```
```   426   "blinfun_of_matrix a x = (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. ((x \<bullet> j) * a i j) *\<^sub>R i)"
```
```   427   by transfer simp
```
```   428
```
```   429 lemma blinfun_of_matrix_minus: "blinfun_of_matrix x - blinfun_of_matrix y = blinfun_of_matrix (x - y)"
```
```   430   by transfer (auto simp: algebra_simps sum_subtractf)
```
```   431
```
```   432 lemma norm_blinfun_of_matrix:
```
```   433   "norm (blinfun_of_matrix a) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>a i j\<bar>)"
```
```   434   apply (rule norm_blinfun_bound)
```
```   435    apply (simp add: sum_nonneg)
```
```   436   apply (simp only: blinfun_of_matrix_apply sum_distrib_right)
```
```   437   apply (rule order_trans[OF norm_sum sum_mono])
```
```   438   apply (rule order_trans[OF norm_sum sum_mono])
```
```   439   apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
```
```   440   done
```
```   441
```
```   442 lemma tendsto_blinfun_of_matrix:
```
```   443   assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F"
```
```   444   shows "((\<lambda>n. blinfun_of_matrix (b n)) \<longlongrightarrow> blinfun_of_matrix a) F"
```
```   445 proof -
```
```   446   have "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> Zfun (\<lambda>x. norm (b x i j - a i j)) F"
```
```   447     using assms unfolding tendsto_Zfun_iff Zfun_norm_iff .
```
```   448   hence "Zfun (\<lambda>x. (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>b x i j - a i j\<bar>)) F"
```
```   449     by (auto intro!: Zfun_sum)
```
```   450   thus ?thesis
```
```   451     unfolding tendsto_Zfun_iff blinfun_of_matrix_minus
```
```   452     by (rule Zfun_le) (auto intro!: order_trans[OF norm_blinfun_of_matrix])
```
```   453 qed
```
```   454
```
```   455 lemma tendsto_componentwise:
```
```   456   fixes a::"'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
```
```   457     and b::"'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   458   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"
```
```   459   apply (subst blinfun_of_matrix_works[of a, symmetric])
```
```   460   apply (subst blinfun_of_matrix_works[of "b x" for x, symmetric, abs_def])
```
```   461   by (rule tendsto_blinfun_of_matrix)
```
```   462
```
```   463 lemma
```
```   464   continuous_blinfun_componentwiseI:
```
```   465   fixes f:: "'b::t2_space \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'c::euclidean_space"
```
```   466   assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> continuous F (\<lambda>x. (f x) j \<bullet> i)"
```
```   467   shows "continuous F f"
```
```   468   using assms by (auto simp: continuous_def intro!: tendsto_componentwise)
```
```   469
```
```   470 lemma
```
```   471   continuous_blinfun_componentwiseI1:
```
```   472   fixes f:: "'b::t2_space \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'c::real_normed_vector"
```
```   473   assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous F (\<lambda>x. f x i)"
```
```   474   shows "continuous F f"
```
```   475   using assms by (auto simp: continuous_def intro!: tendsto_componentwise1)
```
```   476
```
```   477 lemma
```
```   478   continuous_on_blinfun_componentwise:
```
```   479   fixes f:: "'d::t2_space \<Rightarrow> 'e::euclidean_space \<Rightarrow>\<^sub>L 'f::real_normed_vector"
```
```   480   assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on s (\<lambda>x. f x i)"
```
```   481   shows "continuous_on s f"
```
```   482   using assms
```
```   483   by (auto intro!: continuous_at_imp_continuous_on intro!: tendsto_componentwise1
```
```   484     simp: continuous_on_eq_continuous_within continuous_def)
```
```   485
```
```   486 lemma bounded_linear_blinfun_matrix: "bounded_linear (\<lambda>x. (x::_\<Rightarrow>\<^sub>L _) j \<bullet> i)"
```
```   487   by (auto intro!: bounded_linearI' bounded_linear_intros)
```
```   488
```
```   489 lemma continuous_blinfun_matrix:
```
```   490   fixes f:: "'b::t2_space \<Rightarrow> 'a::real_normed_vector \<Rightarrow>\<^sub>L 'c::real_inner"
```
```   491   assumes "continuous F f"
```
```   492   shows "continuous F (\<lambda>x. (f x) j \<bullet> i)"
```
```   493   by (rule bounded_linear.continuous[OF bounded_linear_blinfun_matrix assms])
```
```   494
```
```   495 lemma continuous_on_blinfun_matrix:
```
```   496   fixes f::"'a::t2_space \<Rightarrow> 'b::real_normed_vector \<Rightarrow>\<^sub>L 'c::real_inner"
```
```   497   assumes "continuous_on S f"
```
```   498   shows "continuous_on S (\<lambda>x. (f x) j \<bullet> i)"
```
```   499   using assms
```
```   500   by (auto simp: continuous_on_eq_continuous_within continuous_blinfun_matrix)
```
```   501
```
```   502 lemma continuous_on_blinfun_of_matrix[continuous_intros]:
```
```   503   assumes "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> continuous_on S (\<lambda>s. g s i j)"
```
```   504   shows "continuous_on S (\<lambda>s. blinfun_of_matrix (g s))"
```
```   505   using assms
```
```   506   by (auto simp: continuous_on intro!: tendsto_blinfun_of_matrix)
```
```   507
```
```   508 lemma mult_if_delta:
```
```   509   "(if P then (1::'a::comm_semiring_1) else 0) * q = (if P then q else 0)"
```
```   510   by auto
```
```   511
```
```   512 lemma compact_blinfun_lemma:
```
```   513   fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow>\<^sub>L 'b::euclidean_space"
```
```   514   assumes "bounded (range f)"
```
```   515   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a \<Rightarrow>\<^sub>L 'b. \<exists> r::nat\<Rightarrow>nat.
```
```   516     strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) i) (l i) < e) sequentially)"
```
```   517   by (rule compact_lemma_general[where unproj = "\<lambda>e. blinfun_of_matrix (\<lambda>i j. e j \<bullet> i)"])
```
```   518    (auto intro!: euclidean_eqI[where 'a='b] bounded_linear_image assms
```
```   519     simp: blinfun_of_matrix_works blinfun_of_matrix_apply inner_Basis mult_if_delta sum.delta'
```
```   520       scaleR_sum_left[symmetric])
```
```   521
```
```   522 lemma blinfun_euclidean_eqI: "(\<And>i. i \<in> Basis \<Longrightarrow> blinfun_apply x i = blinfun_apply y i) \<Longrightarrow> x = y"
```
```   523   apply (auto intro!: blinfun_eqI)
```
```   524   apply (subst (2) euclidean_representation[symmetric, where 'a='a])
```
```   525   apply (subst (1) euclidean_representation[symmetric, where 'a='a])
```
```   526   apply (simp add: blinfun.bilinear_simps)
```
```   527   done
```
```   528
```
```   529 lemma Blinfun_eq_matrix: "bounded_linear f \<Longrightarrow> Blinfun f = blinfun_of_matrix (\<lambda>i j. f j \<bullet> i)"
```
```   530   by (intro blinfun_euclidean_eqI)
```
```   531      (auto simp: blinfun_of_matrix_apply bounded_linear_Blinfun_apply inner_Basis if_distrib
```
```   532       if_distribR sum.delta' euclidean_representation
```
```   533       cong: if_cong)
```
```   534
```
```   535 text \<open>TODO: generalize (via \<open>compact_cball\<close>)?\<close>
```
```   536 instance blinfun :: (euclidean_space, euclidean_space) heine_borel
```
```   537 proof
```
```   538   fix f :: "nat \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
```
```   539   assume f: "bounded (range f)"
```
```   540   then obtain l::"'a \<Rightarrow>\<^sub>L 'b" and r where r: "strict_mono r"
```
```   541     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) i) (l i) < e) sequentially"
```
```   542     using compact_blinfun_lemma [OF f] by blast
```
```   543   {
```
```   544     fix e::real
```
```   545     let ?d = "real_of_nat DIM('a) * real_of_nat DIM('b)"
```
```   546     assume "e > 0"
```
```   547     hence "e / ?d > 0" by (simp add: DIM_positive)
```
```   548     with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) i) (l i) < e / ?d) sequentially"
```
```   549       by simp
```
```   550     moreover
```
```   551     {
```
```   552       fix n
```
```   553       assume n: "\<forall>i\<in>Basis. dist (f (r n) i) (l i) < e / ?d"
```
```   554       have "norm (f (r n) - l) = norm (blinfun_of_matrix (\<lambda>i j. (f (r n) - l) j \<bullet> i))"
```
```   555         unfolding blinfun_of_matrix_works ..
```
```   556       also note norm_blinfun_of_matrix
```
```   557       also have "(\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) <
```
```   558         (\<Sum>i\<in>(Basis::'b set). e / real_of_nat DIM('b))"
```
```   559       proof (rule sum_strict_mono)
```
```   560         fix i::'b assume i: "i \<in> Basis"
```
```   561         have "(\<Sum>j::'a\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) < (\<Sum>j::'a\<in>Basis. e / ?d)"
```
```   562         proof (rule sum_strict_mono)
```
```   563           fix j::'a assume j: "j \<in> Basis"
```
```   564           have "\<bar>(f (r n) - l) j \<bullet> i\<bar> \<le> norm ((f (r n) - l) j)"
```
```   565             by (simp add: Basis_le_norm i)
```
```   566           also have "\<dots> < e / ?d"
```
```   567             using n i j by (auto simp: dist_norm blinfun.bilinear_simps)
```
```   568           finally show "\<bar>(f (r n) - l) j \<bullet> i\<bar> < e / ?d" by simp
```
```   569         qed simp_all
```
```   570         also have "\<dots> \<le> e / real_of_nat DIM('b)"
```
```   571           by simp
```
```   572         finally show "(\<Sum>j\<in>Basis. \<bar>(f (r n) - l) j \<bullet> i\<bar>) < e / real_of_nat DIM('b)"
```
```   573           by simp
```
```   574       qed simp_all
```
```   575       also have "\<dots> \<le> e" by simp
```
```   576       finally have "dist (f (r n)) l < e"
```
```   577         by (auto simp: dist_norm)
```
```   578     }
```
```   579     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
```
```   580       using eventually_elim2 by force
```
```   581   }
```
```   582   then have *: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```   583     unfolding o_def tendsto_iff by simp
```
```   584   with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
```
```   585     by auto
```
```   586 qed
```
```   587
```
```   588
```
```   589 subsection%unimportant \<open>concrete bounded linear functions\<close>
```
```   590
```
```   591 lemma transfer_bounded_bilinear_bounded_linearI:
```
```   592   assumes "g = (\<lambda>i x. (blinfun_apply (f i) x))"
```
```   593   shows "bounded_bilinear g = bounded_linear f"
```
```   594 proof
```
```   595   assume "bounded_bilinear g"
```
```   596   then interpret bounded_bilinear f by (simp add: assms)
```
```   597   show "bounded_linear f"
```
```   598   proof (unfold_locales, safe intro!: blinfun_eqI)
```
```   599     fix i
```
```   600     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
```
```   601       by (auto intro!: blinfun_eqI simp: blinfun.bilinear_simps)
```
```   602     from _ nonneg_bounded show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   603       by (rule ex_reg) (auto intro!: onorm_bound simp: norm_blinfun.rep_eq ac_simps)
```
```   604   qed
```
```   605 qed (auto simp: assms intro!: blinfun.comp)
```
```   606
```
```   607 lemma transfer_bounded_bilinear_bounded_linear[transfer_rule]:
```
```   608   "(rel_fun (rel_fun (=) (pcr_blinfun (=) (=))) (=)) bounded_bilinear bounded_linear"
```
```   609   by (auto simp: pcr_blinfun_def cr_blinfun_def rel_fun_def OO_def
```
```   610     intro!: transfer_bounded_bilinear_bounded_linearI)
```
```   611
```
```   612 context bounded_bilinear
```
```   613 begin
```
```   614
```
```   615 lift_definition prod_left::"'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'c" is "(\<lambda>b a. prod a b)"
```
```   616   by (rule bounded_linear_left)
```
```   617 declare prod_left.rep_eq[simp]
```
```   618
```
```   619 lemma bounded_linear_prod_left[bounded_linear]: "bounded_linear prod_left"
```
```   620   by transfer (rule flip)
```
```   621
```
```   622 lift_definition prod_right::"'a \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c" is "(\<lambda>a b. prod a b)"
```
```   623   by (rule bounded_linear_right)
```
```   624 declare prod_right.rep_eq[simp]
```
```   625
```
```   626 lemma bounded_linear_prod_right[bounded_linear]: "bounded_linear prod_right"
```
```   627   by transfer (rule bounded_bilinear_axioms)
```
```   628
```
```   629 end
```
```   630
```
```   631 lift_definition id_blinfun::"'a::real_normed_vector \<Rightarrow>\<^sub>L 'a" is "\<lambda>x. x"
```
```   632   by (rule bounded_linear_ident)
```
```   633
```
```   634 lemmas blinfun_apply_id_blinfun[simp] = id_blinfun.rep_eq
```
```   635
```
```   636 lemma norm_blinfun_id[simp]:
```
```   637   "norm (id_blinfun::'a::{real_normed_vector, perfect_space} \<Rightarrow>\<^sub>L 'a) = 1"
```
```   638   by transfer (auto simp: onorm_id)
```
```   639
```
```   640 lemma norm_blinfun_id_le:
```
```   641   "norm (id_blinfun::'a::real_normed_vector \<Rightarrow>\<^sub>L 'a) \<le> 1"
```
```   642   by transfer (auto simp: onorm_id_le)
```
```   643
```
```   644
```
```   645 lift_definition fst_blinfun::"('a::real_normed_vector \<times> 'b::real_normed_vector) \<Rightarrow>\<^sub>L 'a" is fst
```
```   646   by (rule bounded_linear_fst)
```
```   647
```
```   648 lemma blinfun_apply_fst_blinfun[simp]: "blinfun_apply fst_blinfun = fst"
```
```   649   by transfer (rule refl)
```
```   650
```
```   651
```
```   652 lift_definition snd_blinfun::"('a::real_normed_vector \<times> 'b::real_normed_vector) \<Rightarrow>\<^sub>L 'b" is snd
```
```   653   by (rule bounded_linear_snd)
```
```   654
```
```   655 lemma blinfun_apply_snd_blinfun[simp]: "blinfun_apply snd_blinfun = snd"
```
```   656   by transfer (rule refl)
```
```   657
```
```   658
```
```   659 lift_definition blinfun_compose::
```
```   660   "'a::real_normed_vector \<Rightarrow>\<^sub>L 'b::real_normed_vector \<Rightarrow>
```
```   661     'c::real_normed_vector \<Rightarrow>\<^sub>L 'a \<Rightarrow>
```
```   662     'c \<Rightarrow>\<^sub>L 'b" (infixl "o\<^sub>L" 55) is "(o)"
```
```   663   parametric comp_transfer
```
```   664   unfolding o_def
```
```   665   by (rule bounded_linear_compose)
```
```   666
```
```   667 lemma blinfun_apply_blinfun_compose[simp]: "(a o\<^sub>L b) c = a (b c)"
```
```   668   by (simp add: blinfun_compose.rep_eq)
```
```   669
```
```   670 lemma norm_blinfun_compose:
```
```   671   "norm (f o\<^sub>L g) \<le> norm f * norm g"
```
```   672   by transfer (rule onorm_compose)
```
```   673
```
```   674 lemma bounded_bilinear_blinfun_compose[bounded_bilinear]: "bounded_bilinear (o\<^sub>L)"
```
```   675   by unfold_locales
```
```   676     (auto intro!: blinfun_eqI exI[where x=1] simp: blinfun.bilinear_simps norm_blinfun_compose)
```
```   677
```
```   678 lemma blinfun_compose_zero[simp]:
```
```   679   "blinfun_compose 0 = (\<lambda>_. 0)"
```
```   680   "blinfun_compose x 0 = 0"
```
```   681   by (auto simp: blinfun.bilinear_simps intro!: blinfun_eqI)
```
```   682
```
```   683
```
```   684 lift_definition blinfun_inner_right::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "(\<bullet>)"
```
```   685   by (rule bounded_linear_inner_right)
```
```   686 declare blinfun_inner_right.rep_eq[simp]
```
```   687
```
```   688 lemma bounded_linear_blinfun_inner_right[bounded_linear]: "bounded_linear blinfun_inner_right"
```
```   689   by transfer (rule bounded_bilinear_inner)
```
```   690
```
```   691
```
```   692 lift_definition blinfun_inner_left::"'a::real_inner \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" is "\<lambda>x y. y \<bullet> x"
```
```   693   by (rule bounded_linear_inner_left)
```
```   694 declare blinfun_inner_left.rep_eq[simp]
```
```   695
```
```   696 lemma bounded_linear_blinfun_inner_left[bounded_linear]: "bounded_linear blinfun_inner_left"
```
```   697   by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_inner])
```
```   698
```
```   699
```
```   700 lift_definition blinfun_scaleR_right::"real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_vector" is "(*\<^sub>R)"
```
```   701   by (rule bounded_linear_scaleR_right)
```
```   702 declare blinfun_scaleR_right.rep_eq[simp]
```
```   703
```
```   704 lemma bounded_linear_blinfun_scaleR_right[bounded_linear]: "bounded_linear blinfun_scaleR_right"
```
```   705   by transfer (rule bounded_bilinear_scaleR)
```
```   706
```
```   707
```
```   708 lift_definition blinfun_scaleR_left::"'a::real_normed_vector \<Rightarrow> real \<Rightarrow>\<^sub>L 'a" is "\<lambda>x y. y *\<^sub>R x"
```
```   709   by (rule bounded_linear_scaleR_left)
```
```   710 lemmas [simp] = blinfun_scaleR_left.rep_eq
```
```   711
```
```   712 lemma bounded_linear_blinfun_scaleR_left[bounded_linear]: "bounded_linear blinfun_scaleR_left"
```
```   713   by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_scaleR])
```
```   714
```
```   715
```
```   716 lift_definition blinfun_mult_right::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a::real_normed_algebra" is "(*)"
```
```   717   by (rule bounded_linear_mult_right)
```
```   718 declare blinfun_mult_right.rep_eq[simp]
```
```   719
```
```   720 lemma bounded_linear_blinfun_mult_right[bounded_linear]: "bounded_linear blinfun_mult_right"
```
```   721   by transfer (rule bounded_bilinear_mult)
```
```   722
```
```   723
```
```   724 lift_definition blinfun_mult_left::"'a::real_normed_algebra \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a" is "\<lambda>x y. y * x"
```
```   725   by (rule bounded_linear_mult_left)
```
```   726 lemmas [simp] = blinfun_mult_left.rep_eq
```
```   727
```
```   728 lemma bounded_linear_blinfun_mult_left[bounded_linear]: "bounded_linear blinfun_mult_left"
```
```   729   by transfer (rule bounded_bilinear.flip[OF bounded_bilinear_mult])
```
```   730
```
```   731 lemmas bounded_linear_function_uniform_limit_intros[uniform_limit_intros] =
```
```   732   bounded_linear.uniform_limit[OF bounded_linear_apply_blinfun]
```
```   733   bounded_linear.uniform_limit[OF bounded_linear_blinfun_apply]
```
```   734   bounded_linear.uniform_limit[OF bounded_linear_blinfun_matrix]
```
```   735
```
```   736 end
```