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src/HOL/Hahn_Banach/Subspace.thy

author | paulson <lp15@cam.ac.uk> |

Mon May 23 15:33:24 2016 +0100 (2016-05-23) | |

changeset 63114 | 27afe7af7379 |

parent 63040 | eb4ddd18d635 |

child 63910 | e4fdf9580372 |

permissions | -rw-r--r-- |

Lots of new material for multivariate analysis

1 (* Title: HOL/Hahn_Banach/Subspace.thy

2 Author: Gertrud Bauer, TU Munich

3 *)

5 section \<open>Subspaces\<close>

7 theory Subspace

8 imports Vector_Space "~~/src/HOL/Library/Set_Algebras"

9 begin

11 subsection \<open>Definition\<close>

13 text \<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>

18 locale 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"

25 notation (symbols)

26 subspace (infix "\<unlhd>" 50)

28 declare vectorspace.intro [intro?] subspace.intro [intro?]

30 lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"

31 by (rule subspace.subset)

33 lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"

34 using subset by blast

36 lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"

37 by (rule subspace.subsetD)

39 lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"

40 by (rule subspace.subsetD)

42 lemma (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"

46 proof -

47 interpret vectorspace V by fact

48 from x y show ?thesis by (simp add: diff_eq1 negate_eq1)

49 qed

51 text \<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>

58 lemma (in subspace) zero [intro]:

59 assumes "vectorspace V"

60 shows "0 \<in> U"

61 proof -

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 .

68 qed

70 lemma (in subspace) neg_closed [iff]:

71 assumes "vectorspace V"

72 assumes x: "x \<in> U"

73 shows "- x \<in> U"

74 proof -

75 interpret vectorspace V by fact

76 from x show ?thesis by (simp add: negate_eq1)

77 qed

79 text \<open>\<^medskip> Further derived laws: every subspace is a vector space.\<close>

81 lemma (in subspace) vectorspace [iff]:

82 assumes "vectorspace V"

83 shows "vectorspace U"

84 proof -

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

104 qed

107 text \<open>The subspace relation is reflexive.\<close>

109 lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"

110 proof

111 show "V \<noteq> {}" ..

112 show "V \<subseteq> V" ..

113 next

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

118 qed

120 text \<open>The subspace relation is transitive.\<close>

122 lemma (in vectorspace) subspace_trans [trans]:

123 "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"

124 proof

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)

136 qed

139 subsection \<open>Linear closure\<close>

141 text \<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>

146 definition lin :: "('a::{minus,plus,zero}) \<Rightarrow> 'a set"

147 where "lin x = {a \<cdot> x | a. True}"

149 lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"

150 unfolding lin_def by blast

152 lemma linI' [iff]: "a \<cdot> x \<in> lin x"

153 unfolding lin_def by blast

155 lemma 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

161 text \<open>Every vector is contained in its linear closure.\<close>

163 lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"

164 proof -

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 .

169 qed

171 lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"

172 proof

173 assume "x \<in> V"

174 then show "0 = 0 \<cdot> x" by simp

175 qed

177 text \<open>Any linear closure is a subspace.\<close>

179 lemma (in vectorspace) lin_subspace [intro]:

180 assumes x: "x \<in> V"

181 shows "lin x \<unlhd> V"

182 proof

183 from x show "lin x \<noteq> {}" by auto

184 next

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

191 next

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

210 qed

213 text \<open>Any linear closure is a vector space.\<close>

215 lemma (in vectorspace) lin_vectorspace [intro]:

216 assumes "x \<in> V"

217 shows "vectorspace (lin x)"

218 proof -

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)

223 qed

226 subsection \<open>Sum of two vectorspaces\<close>

228 text \<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>

233 lemma sum_def: "U + V = {u + v | u v. u \<in> U \<and> v \<in> V}"

234 unfolding set_plus_def by auto

236 lemma 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

240 lemma 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

244 lemma sumI' [intro]:

245 "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"

246 unfolding sum_def by blast

248 text \<open>\<open>U\<close> is a subspace of \<open>U + V\<close>.\<close>

250 lemma subspace_sum1 [iff]:

251 assumes "vectorspace U" "vectorspace V"

252 shows "U \<unlhd> U + V"

253 proof -

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

270 qed

272 text \<open>The sum of two subspaces is again a subspace.\<close>

274 lemma sum_subspace [intro?]:

275 assumes "subspace U E" "vectorspace E" "subspace V E"

276 shows "U + V \<unlhd> E"

277 proof -

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

319 qed

321 text \<open>The sum of two subspaces is a vectorspace.\<close>

323 lemma 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)

328 subsection \<open>Direct sums\<close>

330 text \<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>

337 lemma 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"

344 proof -

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:)

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

376 qed

378 text \<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>

385 lemma 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"

391 proof -

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

433 qed

435 text \<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>

441 lemma 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)"

446 proof -

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

460 qed

462 text \<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>

467 lemma 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"

478 proof -

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>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (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)

505 qed

507 end