src/HOL/Analysis/Linear_Algebra.thy
 author nipkow Sat Dec 29 15:43:53 2018 +0100 (6 months ago) changeset 69529 4ab9657b3257 parent 69517 dc20f278e8f3 child 69597 ff784d5a5bfb permissions -rw-r--r--
capitalize proper names in lemma names
```     1 (*  Title:      HOL/Analysis/Linear_Algebra.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section \<open>Elementary Linear Algebra on Euclidean Spaces\<close>
```
```     6
```
```     7 theory Linear_Algebra
```
```     8 imports
```
```     9   Euclidean_Space
```
```    10   "HOL-Library.Infinite_Set"
```
```    11 begin
```
```    12
```
```    13 lemma linear_simps:
```
```    14   assumes "bounded_linear f"
```
```    15   shows
```
```    16     "f (a + b) = f a + f b"
```
```    17     "f (a - b) = f a - f b"
```
```    18     "f 0 = 0"
```
```    19     "f (- a) = - f a"
```
```    20     "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
```
```    21 proof -
```
```    22   interpret f: bounded_linear f by fact
```
```    23   show "f (a + b) = f a + f b" by (rule f.add)
```
```    24   show "f (a - b) = f a - f b" by (rule f.diff)
```
```    25   show "f 0 = 0" by (rule f.zero)
```
```    26   show "f (- a) = - f a" by (rule f.neg)
```
```    27   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale)
```
```    28 qed
```
```    29
```
```    30 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
```
```    31   using finite finite_image_set by blast
```
```    32
```
```    33
```
```    34 subsection%unimportant \<open>More interesting properties of the norm\<close>
```
```    35
```
```    36 notation inner (infix "\<bullet>" 70)
```
```    37
```
```    38 text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
```
```    39
```
```    40 lemma linear_componentwise:
```
```    41   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
```
```    42   assumes lf: "linear f"
```
```    43   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```    44 proof -
```
```    45   interpret linear f by fact
```
```    46   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```    47     by (simp add: inner_sum_left)
```
```    48   then show ?thesis
```
```    49     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
```
```    50 qed
```
```    51
```
```    52 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
```
```    53   (is "?lhs \<longleftrightarrow> ?rhs")
```
```    54 proof
```
```    55   assume ?lhs
```
```    56   then show ?rhs by simp
```
```    57 next
```
```    58   assume ?rhs
```
```    59   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
```
```    60     by simp
```
```    61   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
```
```    62     by (simp add: inner_diff inner_commute)
```
```    63   then have "(x - y) \<bullet> (x - y) = 0"
```
```    64     by (simp add: field_simps inner_diff inner_commute)
```
```    65   then show "x = y" by simp
```
```    66 qed
```
```    67
```
```    68 lemma norm_triangle_half_r:
```
```    69   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
```
```    70   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
```
```    71
```
```    72 lemma norm_triangle_half_l:
```
```    73   assumes "norm (x - y) < e / 2"
```
```    74     and "norm (x' - y) < e / 2"
```
```    75   shows "norm (x - x') < e"
```
```    76   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
```
```    77   unfolding dist_norm[symmetric] .
```
```    78
```
```    79 lemma abs_triangle_half_r:
```
```    80   fixes y :: "'a::linordered_field"
```
```    81   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
```
```    82   by linarith
```
```    83
```
```    84 lemma abs_triangle_half_l:
```
```    85   fixes y :: "'a::linordered_field"
```
```    86   assumes "abs (x - y) < e / 2"
```
```    87     and "abs (x' - y) < e / 2"
```
```    88   shows "abs (x - x') < e"
```
```    89   using assms by linarith
```
```    90
```
```    91 lemma sum_clauses:
```
```    92   shows "sum f {} = 0"
```
```    93     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
```
```    94   by (auto simp add: insert_absorb)
```
```    95
```
```    96 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
```
```    97 proof
```
```    98   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
```
```    99   then have "\<forall>x. x \<bullet> (y - z) = 0"
```
```   100     by (simp add: inner_diff)
```
```   101   then have "(y - z) \<bullet> (y - z) = 0" ..
```
```   102   then show "y = z" by simp
```
```   103 qed simp
```
```   104
```
```   105 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
```
```   106 proof
```
```   107   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
```
```   108   then have "\<forall>z. (x - y) \<bullet> z = 0"
```
```   109     by (simp add: inner_diff)
```
```   110   then have "(x - y) \<bullet> (x - y) = 0" ..
```
```   111   then show "x = y" by simp
```
```   112 qed simp
```
```   113
```
```   114
```
```   115 subsection \<open>Orthogonality\<close>
```
```   116
```
```   117 definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
```
```   118
```
```   119 context real_inner
```
```   120 begin
```
```   121
```
```   122 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
```
```   123   by (simp add: orthogonal_def)
```
```   124
```
```   125 lemma orthogonal_clauses:
```
```   126   "orthogonal a 0"
```
```   127   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
```
```   128   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
```
```   129   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
```
```   130   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
```
```   131   "orthogonal 0 a"
```
```   132   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
```
```   133   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
```
```   134   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
```
```   135   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
```
```   136   unfolding orthogonal_def inner_add inner_diff by auto
```
```   137
```
```   138 end
```
```   139
```
```   140 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
```
```   141   by (simp add: orthogonal_def inner_commute)
```
```   142
```
```   143 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
```
```   144   by (rule ext) (simp add: orthogonal_def)
```
```   145
```
```   146 lemma pairwise_ortho_scaleR:
```
```   147     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
```
```   148     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
```
```   149   by (auto simp: pairwise_def orthogonal_clauses)
```
```   150
```
```   151 lemma orthogonal_rvsum:
```
```   152     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
```
```   153   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   154
```
```   155 lemma orthogonal_lvsum:
```
```   156     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
```
```   157   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
```
```   158
```
```   159 lemma norm_add_Pythagorean:
```
```   160   assumes "orthogonal a b"
```
```   161     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
```
```   162 proof -
```
```   163   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
```
```   164     by (simp add: algebra_simps orthogonal_def inner_commute)
```
```   165   then show ?thesis
```
```   166     by (simp add: power2_norm_eq_inner)
```
```   167 qed
```
```   168
```
```   169 lemma norm_sum_Pythagorean:
```
```   170   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
```
```   171     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
```
```   172 using assms
```
```   173 proof (induction I rule: finite_induct)
```
```   174   case empty then show ?case by simp
```
```   175 next
```
```   176   case (insert x I)
```
```   177   then have "orthogonal (f x) (sum f I)"
```
```   178     by (metis pairwise_insert orthogonal_rvsum)
```
```   179   with insert show ?case
```
```   180     by (simp add: pairwise_insert norm_add_Pythagorean)
```
```   181 qed
```
```   182
```
```   183
```
```   184 subsection \<open>Bilinear functions\<close>
```
```   185
```
```   186 definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
```
```   187
```
```   188 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
```
```   189   by (simp add: bilinear_def linear_iff)
```
```   190
```
```   191 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
```
```   192   by (simp add: bilinear_def linear_iff)
```
```   193
```
```   194 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
```
```   195   by (simp add: bilinear_def linear_iff)
```
```   196
```
```   197 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
```
```   198   by (simp add: bilinear_def linear_iff)
```
```   199
```
```   200 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
```
```   201   by (drule bilinear_lmul [of _ "- 1"]) simp
```
```   202
```
```   203 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
```
```   204   by (drule bilinear_rmul [of _ _ "- 1"]) simp
```
```   205
```
```   206 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
```
```   207   using add_left_imp_eq[of x y 0] by auto
```
```   208
```
```   209 lemma bilinear_lzero:
```
```   210   assumes "bilinear h"
```
```   211   shows "h 0 x = 0"
```
```   212   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
```
```   213
```
```   214 lemma bilinear_rzero:
```
```   215   assumes "bilinear h"
```
```   216   shows "h x 0 = 0"
```
```   217   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
```
```   218
```
```   219 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
```
```   220   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
```
```   221
```
```   222 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
```
```   223   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
```
```   224
```
```   225 lemma bilinear_sum:
```
```   226   assumes "bilinear h"
```
```   227   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
```
```   228 proof -
```
```   229   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
```
```   230   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
```
```   231   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
```
```   232     by (simp add: l.sum)
```
```   233   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
```
```   234     by (rule sum.cong) (simp_all add: r.sum)
```
```   235   finally show ?thesis
```
```   236     unfolding sum.cartesian_product .
```
```   237 qed
```
```   238
```
```   239
```
```   240 subsection \<open>Adjoints\<close>
```
```   241
```
```   242 definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
```
```   243
```
```   244 lemma adjoint_unique:
```
```   245   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   246   shows "adjoint f = g"
```
```   247   unfolding adjoint_def
```
```   248 proof (rule some_equality)
```
```   249   show "\<forall>x y. inner (f x) y = inner x (g y)"
```
```   250     by (rule assms)
```
```   251 next
```
```   252   fix h
```
```   253   assume "\<forall>x y. inner (f x) y = inner x (h y)"
```
```   254   then have "\<forall>x y. inner x (g y) = inner x (h y)"
```
```   255     using assms by simp
```
```   256   then have "\<forall>x y. inner x (g y - h y) = 0"
```
```   257     by (simp add: inner_diff_right)
```
```   258   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
```
```   259     by simp
```
```   260   then have "\<forall>y. h y = g y"
```
```   261     by simp
```
```   262   then show "h = g" by (simp add: ext)
```
```   263 qed
```
```   264
```
```   265 text \<open>TODO: The following lemmas about adjoints should hold for any
```
```   266   Hilbert space (i.e. complete inner product space).
```
```   267   (see \<^url>\<open>https://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
```
```   268 \<close>
```
```   269
```
```   270 lemma adjoint_works:
```
```   271   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   272   assumes lf: "linear f"
```
```   273   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   274 proof -
```
```   275   interpret linear f by fact
```
```   276   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
```
```   277   proof (intro allI exI)
```
```   278     fix y :: "'m" and x
```
```   279     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
```
```   280     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
```
```   281       by (simp add: euclidean_representation)
```
```   282     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
```
```   283       by (simp add: sum scale)
```
```   284     finally show "f x \<bullet> y = x \<bullet> ?w"
```
```   285       by (simp add: inner_sum_left inner_sum_right mult.commute)
```
```   286   qed
```
```   287   then show ?thesis
```
```   288     unfolding adjoint_def choice_iff
```
```   289     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
```
```   290 qed
```
```   291
```
```   292 lemma adjoint_clauses:
```
```   293   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   294   assumes lf: "linear f"
```
```   295   shows "x \<bullet> adjoint f y = f x \<bullet> y"
```
```   296     and "adjoint f y \<bullet> x = y \<bullet> f x"
```
```   297   by (simp_all add: adjoint_works[OF lf] inner_commute)
```
```   298
```
```   299 lemma adjoint_linear:
```
```   300   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   301   assumes lf: "linear f"
```
```   302   shows "linear (adjoint f)"
```
```   303   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
```
```   304     adjoint_clauses[OF lf] inner_distrib)
```
```   305
```
```   306 lemma adjoint_adjoint:
```
```   307   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   308   assumes lf: "linear f"
```
```   309   shows "adjoint (adjoint f) = f"
```
```   310   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
```
```   311
```
```   312
```
```   313 subsection \<open>Archimedean properties and useful consequences\<close>
```
```   314
```
```   315 text\<open>Bernoulli's inequality\<close>
```
```   316 proposition Bernoulli_inequality:
```
```   317   fixes x :: real
```
```   318   assumes "-1 \<le> x"
```
```   319     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   320 proof (induct n)
```
```   321   case 0
```
```   322   then show ?case by simp
```
```   323 next
```
```   324   case (Suc n)
```
```   325   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
```
```   326     by (simp add: algebra_simps)
```
```   327   also have "... = (1 + x) * (1 + n*x)"
```
```   328     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
```
```   329   also have "... \<le> (1 + x) ^ Suc n"
```
```   330     using Suc.hyps assms mult_left_mono by fastforce
```
```   331   finally show ?case .
```
```   332 qed
```
```   333
```
```   334 corollary Bernoulli_inequality_even:
```
```   335   fixes x :: real
```
```   336   assumes "even n"
```
```   337     shows "1 + n * x \<le> (1 + x) ^ n"
```
```   338 proof (cases "-1 \<le> x \<or> n=0")
```
```   339   case True
```
```   340   then show ?thesis
```
```   341     by (auto simp: Bernoulli_inequality)
```
```   342 next
```
```   343   case False
```
```   344   then have "real n \<ge> 1"
```
```   345     by simp
```
```   346   with False have "n * x \<le> -1"
```
```   347     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
```
```   348   then have "1 + n * x \<le> 0"
```
```   349     by auto
```
```   350   also have "... \<le> (1 + x) ^ n"
```
```   351     using assms
```
```   352     using zero_le_even_power by blast
```
```   353   finally show ?thesis .
```
```   354 qed
```
```   355
```
```   356 corollary real_arch_pow:
```
```   357   fixes x :: real
```
```   358   assumes x: "1 < x"
```
```   359   shows "\<exists>n. y < x^n"
```
```   360 proof -
```
```   361   from x have x0: "x - 1 > 0"
```
```   362     by arith
```
```   363   from reals_Archimedean3[OF x0, rule_format, of y]
```
```   364   obtain n :: nat where n: "y < real n * (x - 1)" by metis
```
```   365   from x0 have x00: "x- 1 \<ge> -1" by arith
```
```   366   from Bernoulli_inequality[OF x00, of n] n
```
```   367   have "y < x^n" by auto
```
```   368   then show ?thesis by metis
```
```   369 qed
```
```   370
```
```   371 corollary real_arch_pow_inv:
```
```   372   fixes x y :: real
```
```   373   assumes y: "y > 0"
```
```   374     and x1: "x < 1"
```
```   375   shows "\<exists>n. x^n < y"
```
```   376 proof (cases "x > 0")
```
```   377   case True
```
```   378   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
```
```   379   from real_arch_pow[OF ix, of "1/y"]
```
```   380   obtain n where n: "1/y < (1/x)^n" by blast
```
```   381   then show ?thesis using y \<open>x > 0\<close>
```
```   382     by (auto simp add: field_simps)
```
```   383 next
```
```   384   case False
```
```   385   with y x1 show ?thesis
```
```   386     by (metis less_le_trans not_less power_one_right)
```
```   387 qed
```
```   388
```
```   389 lemma forall_pos_mono:
```
```   390   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   391     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
```
```   392   by (metis real_arch_inverse)
```
```   393
```
```   394 lemma forall_pos_mono_1:
```
```   395   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
```
```   396     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
```
```   397   apply (rule forall_pos_mono)
```
```   398   apply auto
```
```   399   apply (metis Suc_pred of_nat_Suc)
```
```   400   done
```
```   401
```
```   402
```
```   403 subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
```
```   404
```
```   405 lemma independent_Basis: "independent Basis"
```
```   406   by (rule independent_Basis)
```
```   407
```
```   408 lemma span_Basis [simp]: "span Basis = UNIV"
```
```   409   by (rule span_Basis)
```
```   410
```
```   411 lemma in_span_Basis: "x \<in> span Basis"
```
```   412   unfolding span_Basis ..
```
```   413
```
```   414
```
```   415 subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
```
```   416
```
```   417 lemma linear_bounded:
```
```   418   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   419   assumes lf: "linear f"
```
```   420   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   421 proof
```
```   422   interpret linear f by fact
```
```   423   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
```
```   424   show "\<forall>x. norm (f x) \<le> ?B * norm x"
```
```   425   proof
```
```   426     fix x :: 'a
```
```   427     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
```
```   428     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
```
```   429       unfolding euclidean_representation ..
```
```   430     also have "\<dots> = norm (sum ?g Basis)"
```
```   431       by (simp add: sum scale)
```
```   432     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
```
```   433     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
```
```   434     proof -
```
```   435       from Basis_le_norm[OF that, of x]
```
```   436       show "norm (?g i) \<le> norm (f i) * norm x"
```
```   437         unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
```
```   438     qed
```
```   439     from sum_norm_le[of _ ?g, OF th]
```
```   440     show "norm (f x) \<le> ?B * norm x"
```
```   441       unfolding th0 sum_distrib_right by metis
```
```   442   qed
```
```   443 qed
```
```   444
```
```   445 lemma linear_conv_bounded_linear:
```
```   446   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   447   shows "linear f \<longleftrightarrow> bounded_linear f"
```
```   448 proof
```
```   449   assume "linear f"
```
```   450   then interpret f: linear f .
```
```   451   show "bounded_linear f"
```
```   452   proof
```
```   453     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
```
```   454       using \<open>linear f\<close> by (rule linear_bounded)
```
```   455     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   456       by (simp add: mult.commute)
```
```   457   qed
```
```   458 next
```
```   459   assume "bounded_linear f"
```
```   460   then interpret f: bounded_linear f .
```
```   461   show "linear f" ..
```
```   462 qed
```
```   463
```
```   464 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
```
```   465
```
```   466 lemma linear_bounded_pos:
```
```   467   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   468   assumes lf: "linear f"
```
```   469  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
```
```   470 proof -
```
```   471   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
```
```   472     using lf unfolding linear_conv_bounded_linear
```
```   473     by (rule bounded_linear.pos_bounded)
```
```   474   with that show ?thesis
```
```   475     by (auto simp: mult.commute)
```
```   476 qed
```
```   477
```
```   478 lemma linear_invertible_bounded_below_pos:
```
```   479   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   480   assumes "linear f" "linear g" "g \<circ> f = id"
```
```   481   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   482 proof -
```
```   483   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
```
```   484     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
```
```   485   show thesis
```
```   486   proof
```
```   487     show "0 < 1/B"
```
```   488       by (simp add: \<open>B > 0\<close>)
```
```   489     show "1/B * norm x \<le> norm (f x)" for x
```
```   490     proof -
```
```   491       have "1/B * norm x = 1/B * norm (g (f x))"
```
```   492         using assms by (simp add: pointfree_idE)
```
```   493       also have "\<dots> \<le> norm (f x)"
```
```   494         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
```
```   495       finally show ?thesis .
```
```   496     qed
```
```   497   qed
```
```   498 qed
```
```   499
```
```   500 lemma linear_inj_bounded_below_pos:
```
```   501   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
```
```   502   assumes "linear f" "inj f"
```
```   503   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
```
```   504   using linear_injective_left_inverse [OF assms]
```
```   505     linear_invertible_bounded_below_pos assms by blast
```
```   506
```
```   507 lemma bounded_linearI':
```
```   508   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   509   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   510     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```   511   shows "bounded_linear f"
```
```   512   using assms linearI linear_conv_bounded_linear by blast
```
```   513
```
```   514 lemma bilinear_bounded:
```
```   515   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
```
```   516   assumes bh: "bilinear h"
```
```   517   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   518 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
```
```   519   fix x :: 'm
```
```   520   fix y :: 'n
```
```   521   have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
```
```   522     by (simp add: euclidean_representation)
```
```   523   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
```
```   524     unfolding bilinear_sum[OF bh] ..
```
```   525   finally have th: "norm (h x y) = \<dots>" .
```
```   526   have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
```
```   527            \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
```
```   528     by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
```
```   529   then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
```
```   530     unfolding sum_distrib_right th sum.cartesian_product
```
```   531     by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
```
```   532       field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
```
```   533 qed
```
```   534
```
```   535 lemma bilinear_conv_bounded_bilinear:
```
```   536   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   537   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
```
```   538 proof
```
```   539   assume "bilinear h"
```
```   540   show "bounded_bilinear h"
```
```   541   proof
```
```   542     fix x y z
```
```   543     show "h (x + y) z = h x z + h y z"
```
```   544       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   545   next
```
```   546     fix x y z
```
```   547     show "h x (y + z) = h x y + h x z"
```
```   548       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
```
```   549   next
```
```   550     show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
```
```   551       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
```
```   552       by simp_all
```
```   553   next
```
```   554     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   555       using \<open>bilinear h\<close> by (rule bilinear_bounded)
```
```   556     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
```
```   557       by (simp add: ac_simps)
```
```   558   qed
```
```   559 next
```
```   560   assume "bounded_bilinear h"
```
```   561   then interpret h: bounded_bilinear h .
```
```   562   show "bilinear h"
```
```   563     unfolding bilinear_def linear_conv_bounded_linear
```
```   564     using h.bounded_linear_left h.bounded_linear_right by simp
```
```   565 qed
```
```   566
```
```   567 lemma bilinear_bounded_pos:
```
```   568   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
```
```   569   assumes bh: "bilinear h"
```
```   570   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
```
```   571 proof -
```
```   572   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
```
```   573     using bh [unfolded bilinear_conv_bounded_bilinear]
```
```   574     by (rule bounded_bilinear.pos_bounded)
```
```   575   then show ?thesis
```
```   576     by (simp only: ac_simps)
```
```   577 qed
```
```   578
```
```   579 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
```
```   580   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
```
```   581       dest: bounded_linear.linear)
```
```   582
```
```   583 lemma linear_imp_has_derivative:
```
```   584   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   585   shows "linear f \<Longrightarrow> (f has_derivative f) net"
```
```   586   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
```
```   587
```
```   588 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
```
```   589   using bounded_linear_imp_has_derivative differentiable_def by blast
```
```   590
```
```   591 lemma linear_imp_differentiable:
```
```   592   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   593   shows "linear f \<Longrightarrow> f differentiable net"
```
```   594   by (metis linear_imp_has_derivative differentiable_def)
```
```   595
```
```   596
```
```   597 subsection%unimportant \<open>We continue\<close>
```
```   598
```
```   599 lemma independent_bound:
```
```   600   fixes S :: "'a::euclidean_space set"
```
```   601   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
```
```   602   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
```
```   603
```
```   604 lemmas independent_imp_finite = finiteI_independent
```
```   605
```
```   606 corollary
```
```   607   fixes S :: "'a::euclidean_space set"
```
```   608   assumes "independent S"
```
```   609   shows independent_card_le:"card S \<le> DIM('a)"
```
```   610   using assms independent_bound by auto
```
```   611
```
```   612 lemma dependent_biggerset:
```
```   613   fixes S :: "'a::euclidean_space set"
```
```   614   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
```
```   615   by (metis independent_bound not_less)
```
```   616
```
```   617 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
```
```   618
```
```   619 lemma vector_sub_project_orthogonal:
```
```   620   fixes b x :: "'a::euclidean_space"
```
```   621   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
```
```   622   unfolding inner_simps by auto
```
```   623
```
```   624 lemma pairwise_orthogonal_insert:
```
```   625   assumes "pairwise orthogonal S"
```
```   626     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
```
```   627   shows "pairwise orthogonal (insert x S)"
```
```   628   using assms unfolding pairwise_def
```
```   629   by (auto simp add: orthogonal_commute)
```
```   630
```
```   631 lemma basis_orthogonal:
```
```   632   fixes B :: "'a::real_inner set"
```
```   633   assumes fB: "finite B"
```
```   634   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
```
```   635   (is " \<exists>C. ?P B C")
```
```   636   using fB
```
```   637 proof (induct rule: finite_induct)
```
```   638   case empty
```
```   639   then show ?case
```
```   640     apply (rule exI[where x="{}"])
```
```   641     apply (auto simp add: pairwise_def)
```
```   642     done
```
```   643 next
```
```   644   case (insert a B)
```
```   645   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
```
```   646   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
```
```   647   obtain C where C: "finite C" "card C \<le> card B"
```
```   648     "span C = span B" "pairwise orthogonal C" by blast
```
```   649   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
```
```   650   let ?C = "insert ?a C"
```
```   651   from C(1) have fC: "finite ?C"
```
```   652     by simp
```
```   653   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
```
```   654     by (simp add: card_insert_if)
```
```   655   {
```
```   656     fix x k
```
```   657     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
```
```   658       by (simp add: field_simps)
```
```   659     have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
```
```   660       apply (simp only: scaleR_right_diff_distrib th0)
```
```   661       apply (rule span_add_eq)
```
```   662       apply (rule span_scale)
```
```   663       apply (rule span_sum)
```
```   664       apply (rule span_scale)
```
```   665       apply (rule span_base)
```
```   666       apply assumption
```
```   667       done
```
```   668   }
```
```   669   then have SC: "span ?C = span (insert a B)"
```
```   670     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
```
```   671   {
```
```   672     fix y
```
```   673     assume yC: "y \<in> C"
```
```   674     then have Cy: "C = insert y (C - {y})"
```
```   675       by blast
```
```   676     have fth: "finite (C - {y})"
```
```   677       using C by simp
```
```   678     have "orthogonal ?a y"
```
```   679       unfolding orthogonal_def
```
```   680       unfolding inner_diff inner_sum_left right_minus_eq
```
```   681       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
```
```   682       apply (clarsimp simp add: inner_commute[of y a])
```
```   683       apply (rule sum.neutral)
```
```   684       apply clarsimp
```
```   685       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   686       using \<open>y \<in> C\<close> by auto
```
```   687   }
```
```   688   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
```
```   689     by (rule pairwise_orthogonal_insert)
```
```   690   from fC cC SC CPO have "?P (insert a B) ?C"
```
```   691     by blast
```
```   692   then show ?case by blast
```
```   693 qed
```
```   694
```
```   695 lemma orthogonal_basis_exists:
```
```   696   fixes V :: "('a::euclidean_space) set"
```
```   697   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
```
```   698   (card B = dim V) \<and> pairwise orthogonal B"
```
```   699 proof -
```
```   700   from basis_exists[of V] obtain B where
```
```   701     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
```
```   702     by force
```
```   703   from B have fB: "finite B" "card B = dim V"
```
```   704     using independent_bound by auto
```
```   705   from basis_orthogonal[OF fB(1)] obtain C where
```
```   706     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
```
```   707     by blast
```
```   708   from C B have CSV: "C \<subseteq> span V"
```
```   709     by (metis span_superset span_mono subset_trans)
```
```   710   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
```
```   711     by (simp add: span_span)
```
```   712   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
```
```   713   have iC: "independent C"
```
```   714     by (simp add: dim_span)
```
```   715   from C fB have "card C \<le> dim V"
```
```   716     by simp
```
```   717   moreover have "dim V \<le> card C"
```
```   718     using span_card_ge_dim[OF CSV SVC C(1)]
```
```   719     by simp
```
```   720   ultimately have CdV: "card C = dim V"
```
```   721     using C(1) by simp
```
```   722   from C B CSV CdV iC show ?thesis
```
```   723     by auto
```
```   724 qed
```
```   725
```
```   726 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
```
```   727
```
```   728 lemma span_not_univ_orthogonal:
```
```   729   fixes S :: "'a::euclidean_space set"
```
```   730   assumes sU: "span S \<noteq> UNIV"
```
```   731   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
```
```   732 proof -
```
```   733   from sU obtain a where a: "a \<notin> span S"
```
```   734     by blast
```
```   735   from orthogonal_basis_exists obtain B where
```
```   736     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
```
```   737     "card B = dim S" "pairwise orthogonal B"
```
```   738     by blast
```
```   739   from B have fB: "finite B" "card B = dim S"
```
```   740     using independent_bound by auto
```
```   741   from span_mono[OF B(2)] span_mono[OF B(3)]
```
```   742   have sSB: "span S = span B"
```
```   743     by (simp add: span_span)
```
```   744   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
```
```   745   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
```
```   746     unfolding sSB
```
```   747     apply (rule span_sum)
```
```   748     apply (rule span_scale)
```
```   749     apply (rule span_base)
```
```   750     apply assumption
```
```   751     done
```
```   752   with a have a0:"?a  \<noteq> 0"
```
```   753     by auto
```
```   754   have "?a \<bullet> x = 0" if "x\<in>span B" for x
```
```   755   proof (rule span_induct [OF that])
```
```   756     show "subspace {x. ?a \<bullet> x = 0}"
```
```   757       by (auto simp add: subspace_def inner_add)
```
```   758   next
```
```   759     {
```
```   760       fix x
```
```   761       assume x: "x \<in> B"
```
```   762       from x have B': "B = insert x (B - {x})"
```
```   763         by blast
```
```   764       have fth: "finite (B - {x})"
```
```   765         using fB by simp
```
```   766       have "?a \<bullet> x = 0"
```
```   767         apply (subst B')
```
```   768         using fB fth
```
```   769         unfolding sum_clauses(2)[OF fth]
```
```   770         apply simp unfolding inner_simps
```
```   771         apply (clarsimp simp add: inner_add inner_sum_left)
```
```   772         apply (rule sum.neutral, rule ballI)
```
```   773         apply (simp only: inner_commute)
```
```   774         apply (auto simp add: x field_simps
```
```   775           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
```
```   776         done
```
```   777     }
```
```   778     then show "?a \<bullet> x = 0" if "x \<in> B" for x
```
```   779       using that by blast
```
```   780     qed
```
```   781   with a0 show ?thesis
```
```   782     unfolding sSB by (auto intro: exI[where x="?a"])
```
```   783 qed
```
```   784
```
```   785 lemma span_not_univ_subset_hyperplane:
```
```   786   fixes S :: "'a::euclidean_space set"
```
```   787   assumes SU: "span S \<noteq> UNIV"
```
```   788   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   789   using span_not_univ_orthogonal[OF SU] by auto
```
```   790
```
```   791 lemma lowdim_subset_hyperplane:
```
```   792   fixes S :: "'a::euclidean_space set"
```
```   793   assumes d: "dim S < DIM('a)"
```
```   794   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
```
```   795 proof -
```
```   796   {
```
```   797     assume "span S = UNIV"
```
```   798     then have "dim (span S) = dim (UNIV :: ('a) set)"
```
```   799       by simp
```
```   800     then have "dim S = DIM('a)"
```
```   801       by (metis Euclidean_Space.dim_UNIV dim_span)
```
```   802     with d have False by arith
```
```   803   }
```
```   804   then have th: "span S \<noteq> UNIV"
```
```   805     by blast
```
```   806   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
```
```   807 qed
```
```   808
```
```   809 lemma linear_eq_stdbasis:
```
```   810   fixes f :: "'a::euclidean_space \<Rightarrow> _"
```
```   811   assumes lf: "linear f"
```
```   812     and lg: "linear g"
```
```   813     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
```
```   814   shows "f = g"
```
```   815   using linear_eq_on_span[OF lf lg, of Basis] fg
```
```   816   by auto
```
```   817
```
```   818
```
```   819 text \<open>Similar results for bilinear functions.\<close>
```
```   820
```
```   821 lemma bilinear_eq:
```
```   822   assumes bf: "bilinear f"
```
```   823     and bg: "bilinear g"
```
```   824     and SB: "S \<subseteq> span B"
```
```   825     and TC: "T \<subseteq> span C"
```
```   826     and "x\<in>S" "y\<in>T"
```
```   827     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
```
```   828   shows "f x y = g x y"
```
```   829 proof -
```
```   830   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
```
```   831   from bf bg have sp: "subspace ?P"
```
```   832     unfolding bilinear_def linear_iff subspace_def bf bg
```
```   833     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
```
```   834         span_add Ball_def
```
```   835       intro: bilinear_ladd[OF bf])
```
```   836   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
```
```   837     apply (auto simp add: subspace_def)
```
```   838     using bf bg unfolding bilinear_def linear_iff
```
```   839       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
```
```   840         span_add Ball_def
```
```   841       intro: bilinear_ladd[OF bf])
```
```   842     done
```
```   843   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
```
```   844     apply (rule span_induct [OF that sp])
```
```   845     using fg sfg span_induct by blast
```
```   846   then show ?thesis
```
```   847     using SB TC assms by auto
```
```   848 qed
```
```   849
```
```   850 lemma bilinear_eq_stdbasis:
```
```   851   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
```
```   852   assumes bf: "bilinear f"
```
```   853     and bg: "bilinear g"
```
```   854     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
```
```   855   shows "f = g"
```
```   856   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
```
```   857
```
```   858 subsection \<open>Infinity norm\<close>
```
```   859
```
```   860 definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
```
```   861
```
```   862 lemma infnorm_set_image:
```
```   863   fixes x :: "'a::euclidean_space"
```
```   864   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```   865   by blast
```
```   866
```
```   867 lemma infnorm_Max:
```
```   868   fixes x :: "'a::euclidean_space"
```
```   869   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
```
```   870   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
```
```   871
```
```   872 lemma infnorm_set_lemma:
```
```   873   fixes x :: "'a::euclidean_space"
```
```   874   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
```
```   875     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
```
```   876   unfolding infnorm_set_image
```
```   877   by auto
```
```   878
```
```   879 lemma infnorm_pos_le:
```
```   880   fixes x :: "'a::euclidean_space"
```
```   881   shows "0 \<le> infnorm x"
```
```   882   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
```
```   883
```
```   884 lemma infnorm_triangle:
```
```   885   fixes x :: "'a::euclidean_space"
```
```   886   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
```
```   887 proof -
```
```   888   have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
```
```   889     by simp
```
```   890   show ?thesis
```
```   891     by (auto simp: infnorm_Max inner_add_left intro!: *)
```
```   892 qed
```
```   893
```
```   894 lemma infnorm_eq_0:
```
```   895   fixes x :: "'a::euclidean_space"
```
```   896   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
```
```   897 proof -
```
```   898   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
```
```   899     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
```
```   900   then show ?thesis
```
```   901     using infnorm_pos_le[of x] by simp
```
```   902 qed
```
```   903
```
```   904 lemma infnorm_0: "infnorm 0 = 0"
```
```   905   by (simp add: infnorm_eq_0)
```
```   906
```
```   907 lemma infnorm_neg: "infnorm (- x) = infnorm x"
```
```   908   unfolding infnorm_def by simp
```
```   909
```
```   910 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
```
```   911   by (metis infnorm_neg minus_diff_eq)
```
```   912
```
```   913 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
```
```   914 proof -
```
```   915   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
```
```   916     by arith
```
```   917   show ?thesis
```
```   918   proof (rule *)
```
```   919     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
```
```   920     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
```
```   921       by (simp_all add: field_simps infnorm_neg)
```
```   922   qed
```
```   923 qed
```
```   924
```
```   925 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
```
```   926   using infnorm_pos_le[of x] by arith
```
```   927
```
```   928 lemma Basis_le_infnorm:
```
```   929   fixes x :: "'a::euclidean_space"
```
```   930   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
```
```   931   by (simp add: infnorm_Max)
```
```   932
```
```   933 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
```
```   934   unfolding infnorm_Max
```
```   935 proof (safe intro!: Max_eqI)
```
```   936   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
```
```   937   { fix b :: 'a
```
```   938     assume "b \<in> Basis"
```
```   939     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
```
```   940       by (simp add: abs_mult mult_left_mono)
```
```   941   next
```
```   942     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
```
```   943       by (auto simp del: Max_in)
```
```   944     then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
```
```   945       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
```
```   946   }
```
```   947 qed simp
```
```   948
```
```   949 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
```
```   950   unfolding infnorm_mul ..
```
```   951
```
```   952 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
```
```   953   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
```
```   954
```
```   955 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
```
```   956
```
```   957 lemma infnorm_le_norm: "infnorm x \<le> norm x"
```
```   958   by (simp add: Basis_le_norm infnorm_Max)
```
```   959
```
```   960 lemma norm_le_infnorm:
```
```   961   fixes x :: "'a::euclidean_space"
```
```   962   shows "norm x \<le> sqrt DIM('a) * infnorm x"
```
```   963   unfolding norm_eq_sqrt_inner id_def
```
```   964 proof (rule real_le_lsqrt[OF inner_ge_zero])
```
```   965   show "sqrt DIM('a) * infnorm x \<ge> 0"
```
```   966     by (simp add: zero_le_mult_iff infnorm_pos_le)
```
```   967   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
```
```   968     by (metis euclidean_inner order_refl)
```
```   969   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
```
```   970     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
```
```   971   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
```
```   972     by (simp add: power_mult_distrib)
```
```   973   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
```
```   974 qed
```
```   975
```
```   976 lemma tendsto_infnorm [tendsto_intros]:
```
```   977   assumes "(f \<longlongrightarrow> a) F"
```
```   978   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
```
```   979 proof (rule tendsto_compose [OF LIM_I assms])
```
```   980   fix r :: real
```
```   981   assume "r > 0"
```
```   982   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
```
```   983     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
```
```   984 qed
```
```   985
```
```   986 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
```
```   987
```
```   988 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```   989   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   990 proof (cases "x=0")
```
```   991   case True
```
```   992   then show ?thesis
```
```   993     by auto
```
```   994 next
```
```   995   case False
```
```   996   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
```
```   997   have "?rhs \<longleftrightarrow>
```
```   998       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
```
```   999         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
```
```  1000     using False unfolding inner_simps
```
```  1001     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1002   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
```
```  1003     using False  by (simp add: field_simps inner_commute)
```
```  1004   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1005     using False by auto
```
```  1006   finally show ?thesis by metis
```
```  1007 qed
```
```  1008
```
```  1009 lemma norm_cauchy_schwarz_abs_eq:
```
```  1010   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
```
```  1011     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
```
```  1012   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1013 proof -
```
```  1014   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
```
```  1015     by arith
```
```  1016   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
```
```  1017     by simp
```
```  1018   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
```
```  1019     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1020     unfolding norm_minus_cancel norm_scaleR ..
```
```  1021   also have "\<dots> \<longleftrightarrow> ?lhs"
```
```  1022     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
```
```  1023     by auto
```
```  1024   finally show ?thesis ..
```
```  1025 qed
```
```  1026
```
```  1027 lemma norm_triangle_eq:
```
```  1028   fixes x y :: "'a::real_inner"
```
```  1029   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1030 proof (cases "x = 0 \<or> y = 0")
```
```  1031   case True
```
```  1032   then show ?thesis
```
```  1033     by force
```
```  1034 next
```
```  1035   case False
```
```  1036   then have n: "norm x > 0" "norm y > 0"
```
```  1037     by auto
```
```  1038   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
```
```  1039     by simp
```
```  1040   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
```
```  1041     unfolding norm_cauchy_schwarz_eq[symmetric]
```
```  1042     unfolding power2_norm_eq_inner inner_simps
```
```  1043     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
```
```  1044   finally show ?thesis .
```
```  1045 qed
```
```  1046
```
```  1047
```
```  1048 subsection \<open>Collinearity\<close>
```
```  1049
```
```  1050 definition%important collinear :: "'a::real_vector set \<Rightarrow> bool"
```
```  1051   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
```
```  1052
```
```  1053 lemma collinear_alt:
```
```  1054      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
```
```  1055 proof
```
```  1056   assume ?lhs
```
```  1057   then show ?rhs
```
```  1058     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
```
```  1059 next
```
```  1060   assume ?rhs
```
```  1061   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
```
```  1062     by (auto simp: )
```
```  1063   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
```
```  1064         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
```
```  1065   then show ?lhs
```
```  1066     using collinear_def by blast
```
```  1067 qed
```
```  1068
```
```  1069 lemma collinear:
```
```  1070   fixes S :: "'a::{perfect_space,real_vector} set"
```
```  1071   shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))"
```
```  1072 proof -
```
```  1073   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
```
```  1074     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
```
```  1075   proof -
```
```  1076     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
```
```  1077       using that by auto
```
```  1078     moreover
```
```  1079     obtain v::'a where "v \<noteq> 0"
```
```  1080       using UNIV_not_singleton [of 0] by auto
```
```  1081     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
```
```  1082       by auto
```
```  1083     then show ?thesis
```
```  1084       using \<open>v \<noteq> 0\<close> by blast
```
```  1085   qed
```
```  1086   then show ?thesis
```
```  1087     apply (clarsimp simp: collinear_def)
```
```  1088     by (metis scaleR_zero_right vector_fraction_eq_iff)
```
```  1089 qed
```
```  1090
```
```  1091 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
```
```  1092   by (meson collinear_def subsetCE)
```
```  1093
```
```  1094 lemma collinear_empty [iff]: "collinear {}"
```
```  1095   by (simp add: collinear_def)
```
```  1096
```
```  1097 lemma collinear_sing [iff]: "collinear {x}"
```
```  1098   by (simp add: collinear_def)
```
```  1099
```
```  1100 lemma collinear_2 [iff]: "collinear {x, y}"
```
```  1101   apply (simp add: collinear_def)
```
```  1102   apply (rule exI[where x="x - y"])
```
```  1103   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
```
```  1104
```
```  1105 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
```
```  1106   (is "?lhs \<longleftrightarrow> ?rhs")
```
```  1107 proof (cases "x = 0 \<or> y = 0")
```
```  1108   case True
```
```  1109   then show ?thesis
```
```  1110     by (auto simp: insert_commute)
```
```  1111 next
```
```  1112   case False
```
```  1113   show ?thesis
```
```  1114   proof
```
```  1115     assume h: "?lhs"
```
```  1116     then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
```
```  1117       unfolding collinear_def by blast
```
```  1118     from u[rule_format, of x 0] u[rule_format, of y 0]
```
```  1119     obtain cx and cy where
```
```  1120       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
```
```  1121       by auto
```
```  1122     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
```
```  1123     let ?d = "cy / cx"
```
```  1124     from cx cy cx0 have "y = ?d *\<^sub>R x"
```
```  1125       by simp
```
```  1126     then show ?rhs using False by blast
```
```  1127   next
```
```  1128     assume h: "?rhs"
```
```  1129     then obtain c where c: "y = c *\<^sub>R x"
```
```  1130       using False by blast
```
```  1131     show ?lhs
```
```  1132       unfolding collinear_def c
```
```  1133       apply (rule exI[where x=x])
```
```  1134       apply auto
```
```  1135           apply (rule exI[where x="- 1"], simp)
```
```  1136          apply (rule exI[where x= "-c"], simp)
```
```  1137         apply (rule exI[where x=1], simp)
```
```  1138        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
```
```  1139       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
```
```  1140       done
```
```  1141   qed
```
```  1142 qed
```
```  1143
```
```  1144 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
```
```  1145 proof (cases "x=0")
```
```  1146   case True
```
```  1147   then show ?thesis
```
```  1148     by (auto simp: insert_commute)
```
```  1149 next
```
```  1150   case False
```
```  1151   then have nnz: "norm x \<noteq> 0"
```
```  1152     by auto
```
```  1153   show ?thesis
```
```  1154   proof
```
```  1155     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1156     then show "collinear {0, x, y}"
```
```  1157       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
```
```  1158       by (meson eq_vector_fraction_iff nnz)
```
```  1159   next
```
```  1160     assume "collinear {0, x, y}"
```
```  1161     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
```
```  1162       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
```
```  1163   qed
```
```  1164 qed
```
```  1165
```
```  1166 end
```