1(* Title: HOL/Hahn_Banach/Subspace.thy 2 Author: Gertrud Bauer, TU Munich 3*) 4 5section \<open>Subspaces\<close> 6 7theory Subspace 8imports Vector_Space "HOL-Library.Set_Algebras" 9begin 10 11subsection \<open>Definition\<close> 12 13text \<open> 14 A non-empty subset \<open>U\<close> of a vector space \<open>V\<close> is a \<^emph>\<open>subspace\<close> of \<open>V\<close>, iff 15 \<open>U\<close> is closed under addition and scalar multiplication. 16\<close> 17 18locale subspace = 19 fixes U :: "'a::{minus, plus, zero, uminus} set" and V 20 assumes non_empty [iff, intro]: "U \<noteq> {}" 21 and subset [iff]: "U \<subseteq> V" 22 and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U" 23 and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U" 24 25notation (symbols) 26 subspace (infix "\<unlhd>" 50) 27 28declare vectorspace.intro [intro?] subspace.intro [intro?] 29 30lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V" 31 by (rule subspace.subset) 32 33lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V" 34 using subset by blast 35 36lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V" 37 by (rule subspace.subsetD) 38 39lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V" 40 by (rule subspace.subsetD) 41 42lemma (in subspace) diff_closed [iff]: 43 assumes "vectorspace V" 44 assumes x: "x \<in> U" and y: "y \<in> U" 45 shows "x - y \<in> U" 46proof - 47 interpret vectorspace V by fact 48 from x y show ?thesis by (simp add: diff_eq1 negate_eq1) 49qed 50 51text \<open> 52 \<^medskip> 53 Similar as for linear spaces, the existence of the zero element in every 54 subspace follows from the non-emptiness of the carrier set and by vector 55 space laws. 56\<close> 57 58lemma (in subspace) zero [intro]: 59 assumes "vectorspace V" 60 shows "0 \<in> U" 61proof - 62 interpret V: vectorspace V by fact 63 have "U \<noteq> {}" by (rule non_empty) 64 then obtain x where x: "x \<in> U" by blast 65 then have "x \<in> V" .. then have "0 = x - x" by simp 66 also from \<open>vectorspace V\<close> x x have "\<dots> \<in> U" by (rule diff_closed) 67 finally show ?thesis . 68qed 69 70lemma (in subspace) neg_closed [iff]: 71 assumes "vectorspace V" 72 assumes x: "x \<in> U" 73 shows "- x \<in> U" 74proof - 75 interpret vectorspace V by fact 76 from x show ?thesis by (simp add: negate_eq1) 77qed 78 79text \<open>\<^medskip> Further derived laws: every subspace is a vector space.\<close> 80 81lemma (in subspace) vectorspace [iff]: 82 assumes "vectorspace V" 83 shows "vectorspace U" 84proof - 85 interpret vectorspace V by fact 86 show ?thesis 87 proof 88 show "U \<noteq> {}" .. 89 fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U" 90 fix a b :: real 91 from x y show "x + y \<in> U" by simp 92 from x show "a \<cdot> x \<in> U" by simp 93 from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac) 94 from x y show "x + y = y + x" by (simp add: add_ac) 95 from x show "x - x = 0" by simp 96 from x show "0 + x = x" by simp 97 from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib) 98 from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib) 99 from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc) 100 from x show "1 \<cdot> x = x" by simp 101 from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1) 102 from x y show "x - y = x + - y" by (simp add: diff_eq1) 103 qed 104qed 105 106 107text \<open>The subspace relation is reflexive.\<close> 108 109lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V" 110proof 111 show "V \<noteq> {}" .. 112 show "V \<subseteq> V" .. 113next 114 fix x y assume x: "x \<in> V" and y: "y \<in> V" 115 fix a :: real 116 from x y show "x + y \<in> V" by simp 117 from x show "a \<cdot> x \<in> V" by simp 118qed 119 120text \<open>The subspace relation is transitive.\<close> 121 122lemma (in vectorspace) subspace_trans [trans]: 123 "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W" 124proof 125 assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W" 126 from uv show "U \<noteq> {}" by (rule subspace.non_empty) 127 show "U \<subseteq> W" 128 proof - 129 from uv have "U \<subseteq> V" by (rule subspace.subset) 130 also from vw have "V \<subseteq> W" by (rule subspace.subset) 131 finally show ?thesis . 132 qed 133 fix x y assume x: "x \<in> U" and y: "y \<in> U" 134 from uv and x y show "x + y \<in> U" by (rule subspace.add_closed) 135 from uv and x show "a \<cdot> x \<in> U" for a by (rule subspace.mult_closed) 136qed 137 138 139subsection \<open>Linear closure\<close> 140 141text \<open> 142 The \<^emph>\<open>linear closure\<close> of a vector \<open>x\<close> is the set of all scalar multiples of 143 \<open>x\<close>. 144\<close> 145 146definition lin :: "('a::{minus,plus,zero}) \<Rightarrow> 'a set" 147 where "lin x = {a \<cdot> x | a. True}" 148 149lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x" 150 unfolding lin_def by blast 151 152lemma linI' [iff]: "a \<cdot> x \<in> lin x" 153 unfolding lin_def by blast 154 155lemma linE [elim]: 156 assumes "x \<in> lin v" 157 obtains a :: real where "x = a \<cdot> v" 158 using assms unfolding lin_def by blast 159 160 161text \<open>Every vector is contained in its linear closure.\<close> 162 163lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x" 164proof - 165 assume "x \<in> V" 166 then have "x = 1 \<cdot> x" by simp 167 also have "\<dots> \<in> lin x" .. 168 finally show ?thesis . 169qed 170 171lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x" 172proof 173 assume "x \<in> V" 174 then show "0 = 0 \<cdot> x" by simp 175qed 176 177text \<open>Any linear closure is a subspace.\<close> 178 179lemma (in vectorspace) lin_subspace [intro]: 180 assumes x: "x \<in> V" 181 shows "lin x \<unlhd> V" 182proof 183 from x show "lin x \<noteq> {}" by auto 184next 185 show "lin x \<subseteq> V" 186 proof 187 fix x' assume "x' \<in> lin x" 188 then obtain a where "x' = a \<cdot> x" .. 189 with x show "x' \<in> V" by simp 190 qed 191next 192 fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x" 193 show "x' + x'' \<in> lin x" 194 proof - 195 from x' obtain a' where "x' = a' \<cdot> x" .. 196 moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" .. 197 ultimately have "x' + x'' = (a' + a'') \<cdot> x" 198 using x by (simp add: distrib) 199 also have "\<dots> \<in> lin x" .. 200 finally show ?thesis . 201 qed 202 fix a :: real 203 show "a \<cdot> x' \<in> lin x" 204 proof - 205 from x' obtain a' where "x' = a' \<cdot> x" .. 206 with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc) 207 also have "\<dots> \<in> lin x" .. 208 finally show ?thesis . 209 qed 210qed 211 212 213text \<open>Any linear closure is a vector space.\<close> 214 215lemma (in vectorspace) lin_vectorspace [intro]: 216 assumes "x \<in> V" 217 shows "vectorspace (lin x)" 218proof - 219 from \<open>x \<in> V\<close> have "subspace (lin x) V" 220 by (rule lin_subspace) 221 from this and vectorspace_axioms show ?thesis 222 by (rule subspace.vectorspace) 223qed 224 225 226subsection \<open>Sum of two vectorspaces\<close> 227 228text \<open> 229 The \<^emph>\<open>sum\<close> of two vectorspaces \<open>U\<close> and \<open>V\<close> is the set of all sums of 230 elements from \<open>U\<close> and \<open>V\<close>. 231\<close> 232 233lemma sum_def: "U + V = {u + v | u v. u \<in> U \<and> v \<in> V}" 234 unfolding set_plus_def by auto 235 236lemma sumE [elim]: 237 "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C" 238 unfolding sum_def by blast 239 240lemma sumI [intro]: 241 "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V" 242 unfolding sum_def by blast 243 244lemma sumI' [intro]: 245 "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V" 246 unfolding sum_def by blast 247 248text \<open>\<open>U\<close> is a subspace of \<open>U + V\<close>.\<close> 249 250lemma subspace_sum1 [iff]: 251 assumes "vectorspace U" "vectorspace V" 252 shows "U \<unlhd> U + V" 253proof - 254 interpret vectorspace U by fact 255 interpret vectorspace V by fact 256 show ?thesis 257 proof 258 show "U \<noteq> {}" .. 259 show "U \<subseteq> U + V" 260 proof 261 fix x assume x: "x \<in> U" 262 moreover have "0 \<in> V" .. 263 ultimately have "x + 0 \<in> U + V" .. 264 with x show "x \<in> U + V" by simp 265 qed 266 fix x y assume x: "x \<in> U" and "y \<in> U" 267 then show "x + y \<in> U" by simp 268 from x show "a \<cdot> x \<in> U" for a by simp 269 qed 270qed 271 272text \<open>The sum of two subspaces is again a subspace.\<close> 273 274lemma sum_subspace [intro?]: 275 assumes "subspace U E" "vectorspace E" "subspace V E" 276 shows "U + V \<unlhd> E" 277proof - 278 interpret subspace U E by fact 279 interpret vectorspace E by fact 280 interpret subspace V E by fact 281 show ?thesis 282 proof 283 have "0 \<in> U + V" 284 proof 285 show "0 \<in> U" using \<open>vectorspace E\<close> .. 286 show "0 \<in> V" using \<open>vectorspace E\<close> .. 287 show "(0::'a) = 0 + 0" by simp 288 qed 289 then show "U + V \<noteq> {}" by blast 290 show "U + V \<subseteq> E" 291 proof 292 fix x assume "x \<in> U + V" 293 then obtain u v where "x = u + v" and 294 "u \<in> U" and "v \<in> V" .. 295 then show "x \<in> E" by simp 296 qed 297 next 298 fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V" 299 show "x + y \<in> U + V" 300 proof - 301 from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" .. 302 moreover 303 from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" .. 304 ultimately 305 have "ux + uy \<in> U" 306 and "vx + vy \<in> V" 307 and "x + y = (ux + uy) + (vx + vy)" 308 using x y by (simp_all add: add_ac) 309 then show ?thesis .. 310 qed 311 fix a show "a \<cdot> x \<in> U + V" 312 proof - 313 from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" .. 314 then have "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V" 315 and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib) 316 then show ?thesis .. 317 qed 318 qed 319qed 320 321text \<open>The sum of two subspaces is a vectorspace.\<close> 322 323lemma sum_vs [intro?]: 324 "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)" 325 by (rule subspace.vectorspace) (rule sum_subspace) 326 327 328subsection \<open>Direct sums\<close> 329 330text \<open> 331 The sum of \<open>U\<close> and \<open>V\<close> is called \<^emph>\<open>direct\<close>, iff the zero element is the only 332 common element of \<open>U\<close> and \<open>V\<close>. For every element \<open>x\<close> of the direct sum of 333 \<open>U\<close> and \<open>V\<close> the decomposition in \<open>x = u + v\<close> with \<open>u \<in> U\<close> and \<open>v \<in> V\<close> is 334 unique. 335\<close> 336 337lemma decomp: 338 assumes "vectorspace E" "subspace U E" "subspace V E" 339 assumes direct: "U \<inter> V = {0}" 340 and u1: "u1 \<in> U" and u2: "u2 \<in> U" 341 and v1: "v1 \<in> V" and v2: "v2 \<in> V" 342 and sum: "u1 + v1 = u2 + v2" 343 shows "u1 = u2 \<and> v1 = v2" 344proof - 345 interpret vectorspace E by fact 346 interpret subspace U E by fact 347 interpret subspace V E by fact 348 show ?thesis 349 proof 350 have U: "vectorspace U" (* FIXME: use interpret *) 351 using \<open>subspace U E\<close> \<open>vectorspace E\<close> by (rule subspace.vectorspace) 352 have V: "vectorspace V" 353 using \<open>subspace V E\<close> \<open>vectorspace E\<close> by (rule subspace.vectorspace) 354 from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1" 355 by (simp add: add_diff_swap) 356 from u1 u2 have u: "u1 - u2 \<in> U" 357 by (rule vectorspace.diff_closed [OF U]) 358 with eq have v': "v2 - v1 \<in> U" by (simp only:) 359 from v2 v1 have v: "v2 - v1 \<in> V" 360 by (rule vectorspace.diff_closed [OF V]) 361 with eq have u': " u1 - u2 \<in> V" by (simp only:) 362 363 show "u1 = u2" 364 proof (rule add_minus_eq) 365 from u1 show "u1 \<in> E" .. 366 from u2 show "u2 \<in> E" .. 367 from u u' and direct show "u1 - u2 = 0" by blast 368 qed 369 show "v1 = v2" 370 proof (rule add_minus_eq [symmetric]) 371 from v1 show "v1 \<in> E" .. 372 from v2 show "v2 \<in> E" .. 373 from v v' and direct show "v2 - v1 = 0" by blast 374 qed 375 qed 376qed 377 378text \<open> 379 An application of the previous lemma will be used in the proof of the 380 Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any element 381 \<open>y + a \<cdot> x\<^sub>0\<close> of the direct sum of a vectorspace \<open>H\<close> and the linear closure 382 of \<open>x\<^sub>0\<close> the components \<open>y \<in> H\<close> and \<open>a\<close> are uniquely determined. 383\<close> 384 385lemma decomp_H': 386 assumes "vectorspace E" "subspace H E" 387 assumes y1: "y1 \<in> H" and y2: "y2 \<in> H" 388 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 389 and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'" 390 shows "y1 = y2 \<and> a1 = a2" 391proof - 392 interpret vectorspace E by fact 393 interpret subspace H E by fact 394 show ?thesis 395 proof 396 have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'" 397 proof (rule decomp) 398 show "a1 \<cdot> x' \<in> lin x'" .. 399 show "a2 \<cdot> x' \<in> lin x'" .. 400 show "H \<inter> lin x' = {0}" 401 proof 402 show "H \<inter> lin x' \<subseteq> {0}" 403 proof 404 fix x assume x: "x \<in> H \<inter> lin x'" 405 then obtain a where xx': "x = a \<cdot> x'" 406 by blast 407 have "x = 0" 408 proof cases 409 assume "a = 0" 410 with xx' and x' show ?thesis by simp 411 next 412 assume a: "a \<noteq> 0" 413 from x have "x \<in> H" .. 414 with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp 415 with a and x' have "x' \<in> H" by (simp add: mult_assoc2) 416 with \<open>x' \<notin> H\<close> show ?thesis by contradiction 417 qed 418 then show "x \<in> {0}" .. 419 qed 420 show "{0} \<subseteq> H \<inter> lin x'" 421 proof - 422 have "0 \<in> H" using \<open>vectorspace E\<close> .. 423 moreover have "0 \<in> lin x'" using \<open>x' \<in> E\<close> .. 424 ultimately show ?thesis by blast 425 qed 426 qed 427 show "lin x' \<unlhd> E" using \<open>x' \<in> E\<close> .. 428 qed (rule \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, rule y1, rule y2, rule eq) 429 then show "y1 = y2" .. 430 from c have "a1 \<cdot> x' = a2 \<cdot> x'" .. 431 with x' show "a1 = a2" by (simp add: mult_right_cancel) 432 qed 433qed 434 435text \<open> 436 Since for any element \<open>y + a \<cdot> x'\<close> of the direct sum of a vectorspace \<open>H\<close> 437 and the linear closure of \<open>x'\<close> the components \<open>y \<in> H\<close> and \<open>a\<close> are unique, it 438 follows from \<open>y \<in> H\<close> that \<open>a = 0\<close>. 439\<close> 440 441lemma decomp_H'_H: 442 assumes "vectorspace E" "subspace H E" 443 assumes t: "t \<in> H" 444 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 445 shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)" 446proof - 447 interpret vectorspace E by fact 448 interpret subspace H E by fact 449 show ?thesis 450 proof (rule, simp_all only: split_paired_all split_conv) 451 from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp 452 fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H" 453 have "y = t \<and> a = 0" 454 proof (rule decomp_H') 455 from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp 456 from ya show "y \<in> H" .. 457 qed (rule \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, rule t, (rule x')+) 458 with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp 459 qed 460qed 461 462text \<open> 463 The components \<open>y \<in> H\<close> and \<open>a\<close> in \<open>y + a \<cdot> x'\<close> are unique, so the function 464 \<open>h'\<close> defined by \<open>h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>\<close> is definite. 465\<close> 466 467lemma h'_definite: 468 fixes H 469 assumes h'_def: 470 "\<And>x. h' x = 471 (let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H) 472 in (h y) + a * xi)" 473 and x: "x = y + a \<cdot> x'" 474 assumes "vectorspace E" "subspace H E" 475 assumes y: "y \<in> H" 476 and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0" 477 shows "h' x = h y + a * xi" 478proof - 479 interpret vectorspace E by fact 480 interpret subspace H E by fact 481 from x y x' have "x \<in> H + lin x'" by auto 482 have "\<exists>!(y, a). x = y + a \<cdot> x' \<and> y \<in> H" (is "\<exists>!p. ?P p") 483 proof (rule ex_ex1I) 484 from x y show "\<exists>p. ?P p" by blast 485 fix p q assume p: "?P p" and q: "?P q" 486 show "p = q" 487 proof - 488 from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H" 489 by (cases p) simp 490 from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H" 491 by (cases q) simp 492 have "fst p = fst q \<and> snd p = snd q" 493 proof (rule decomp_H') 494 from xp show "fst p \<in> H" .. 495 from xq show "fst q \<in> H" .. 496 from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'" 497 by simp 498 qed (rule \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, (rule x')+) 499 then show ?thesis by (cases p, cases q) simp 500 qed 501 qed 502 then have eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)" 503 by (rule some1_equality) (simp add: x y) 504 with h'_def show "h' x = h y + a * xi" by (simp add: Let_def) 505qed 506 507end 508