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src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy

author | haftmann |

Fri Nov 01 18:51:14 2013 +0100 (2013-11-01) | |

changeset 54230 | b1d955791529 |

parent 52183 | 667961fa6a60 |

child 58744 | c434e37f290e |

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

more simplification rules on unary and binary minus

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

2 Author: Gertrud Bauer, TU Munich

3 *)

5 header {* The supremum w.r.t.~the function order *}

7 theory Hahn_Banach_Sup_Lemmas

8 imports Function_Norm Zorn_Lemma

9 begin

11 text {*

12 This section contains some lemmas that will be used in the proof of

13 the Hahn-Banach Theorem. In this section the following context is

14 presumed. Let @{text E} be a real vector space with a seminorm

15 @{text p} on @{text E}. @{text F} is a subspace of @{text E} and

16 @{text f} a linear form on @{text F}. We consider a chain @{text c}

17 of norm-preserving extensions of @{text f}, such that @{text "\<Union>c =

18 graph H h"}. We will show some properties about the limit function

19 @{text h}, i.e.\ the supremum of the chain @{text c}.

21 \medskip Let @{text c} be a chain of norm-preserving extensions of

22 the function @{text f} and let @{text "graph H h"} be the supremum

23 of @{text c}. Every element in @{text H} is member of one of the

24 elements of the chain.

25 *}

26 lemmas [dest?] = chainsD

27 lemmas chainsE2 [elim?] = chainsD2 [elim_format]

29 lemma some_H'h't:

30 assumes M: "M = norm_pres_extensions E p F f"

31 and cM: "c \<in> chains M"

32 and u: "graph H h = \<Union>c"

33 and x: "x \<in> H"

34 shows "\<exists>H' h'. graph H' h' \<in> c

35 \<and> (x, h x) \<in> graph H' h'

36 \<and> linearform H' h' \<and> H' \<unlhd> E

37 \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h'

38 \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

39 proof -

40 from x have "(x, h x) \<in> graph H h" ..

41 also from u have "\<dots> = \<Union>c" .

42 finally obtain g where gc: "g \<in> c" and gh: "(x, h x) \<in> g" by blast

44 from cM have "c \<subseteq> M" ..

45 with gc have "g \<in> M" ..

46 also from M have "\<dots> = norm_pres_extensions E p F f" .

47 finally obtain H' and h' where g: "g = graph H' h'"

48 and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"

49 "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x" ..

51 from gc and g have "graph H' h' \<in> c" by (simp only:)

52 moreover from gh and g have "(x, h x) \<in> graph H' h'" by (simp only:)

53 ultimately show ?thesis using * by blast

54 qed

56 text {*

57 \medskip Let @{text c} be a chain of norm-preserving extensions of

58 the function @{text f} and let @{text "graph H h"} be the supremum

59 of @{text c}. Every element in the domain @{text H} of the supremum

60 function is member of the domain @{text H'} of some function @{text

61 h'}, such that @{text h} extends @{text h'}.

62 *}

64 lemma some_H'h':

65 assumes M: "M = norm_pres_extensions E p F f"

66 and cM: "c \<in> chains M"

67 and u: "graph H h = \<Union>c"

68 and x: "x \<in> H"

69 shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h

70 \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'

71 \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

72 proof -

73 from M cM u x obtain H' h' where

74 x_hx: "(x, h x) \<in> graph H' h'"

75 and c: "graph H' h' \<in> c"

76 and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"

77 "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"

78 by (rule some_H'h't [elim_format]) blast

79 from x_hx have "x \<in> H'" ..

80 moreover from cM u c have "graph H' h' \<subseteq> graph H h" by blast

81 ultimately show ?thesis using * by blast

82 qed

84 text {*

85 \medskip Any two elements @{text x} and @{text y} in the domain

86 @{text H} of the supremum function @{text h} are both in the domain

87 @{text H'} of some function @{text h'}, such that @{text h} extends

88 @{text h'}.

89 *}

91 lemma some_H'h'2:

92 assumes M: "M = norm_pres_extensions E p F f"

93 and cM: "c \<in> chains M"

94 and u: "graph H h = \<Union>c"

95 and x: "x \<in> H"

96 and y: "y \<in> H"

97 shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H'

98 \<and> graph H' h' \<subseteq> graph H h

99 \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H'

100 \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"

101 proof -

102 txt {* @{text y} is in the domain @{text H''} of some function @{text h''},

103 such that @{text h} extends @{text h''}. *}

105 from M cM u and y obtain H' h' where

106 y_hy: "(y, h y) \<in> graph H' h'"

107 and c': "graph H' h' \<in> c"

108 and * :

109 "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'"

110 "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"

111 by (rule some_H'h't [elim_format]) blast

113 txt {* @{text x} is in the domain @{text H'} of some function @{text h'},

114 such that @{text h} extends @{text h'}. *}

116 from M cM u and x obtain H'' h'' where

117 x_hx: "(x, h x) \<in> graph H'' h''"

118 and c'': "graph H'' h'' \<in> c"

119 and ** :

120 "linearform H'' h''" "H'' \<unlhd> E" "F \<unlhd> H''"

121 "graph F f \<subseteq> graph H'' h''" "\<forall>x \<in> H''. h'' x \<le> p x"

122 by (rule some_H'h't [elim_format]) blast

124 txt {* Since both @{text h'} and @{text h''} are elements of the chain,

125 @{text h''} is an extension of @{text h'} or vice versa. Thus both

126 @{text x} and @{text y} are contained in the greater

127 one. \label{cases1}*}

129 from cM c'' c' have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''"

130 (is "?case1 \<or> ?case2") ..

131 then show ?thesis

132 proof

133 assume ?case1

134 have "(x, h x) \<in> graph H'' h''" by fact

135 also have "\<dots> \<subseteq> graph H' h'" by fact

136 finally have xh:"(x, h x) \<in> graph H' h'" .

137 then have "x \<in> H'" ..

138 moreover from y_hy have "y \<in> H'" ..

139 moreover from cM u and c' have "graph H' h' \<subseteq> graph H h" by blast

140 ultimately show ?thesis using * by blast

141 next

142 assume ?case2

143 from x_hx have "x \<in> H''" ..

144 moreover {

145 have "(y, h y) \<in> graph H' h'" by (rule y_hy)

146 also have "\<dots> \<subseteq> graph H'' h''" by fact

147 finally have "(y, h y) \<in> graph H'' h''" .

148 } then have "y \<in> H''" ..

149 moreover from cM u and c'' have "graph H'' h'' \<subseteq> graph H h" by blast

150 ultimately show ?thesis using ** by blast

151 qed

152 qed

154 text {*

155 \medskip The relation induced by the graph of the supremum of a

156 chain @{text c} is definite, i.~e.~t is the graph of a function.

157 *}

159 lemma sup_definite:

160 assumes M_def: "M \<equiv> norm_pres_extensions E p F f"

161 and cM: "c \<in> chains M"

162 and xy: "(x, y) \<in> \<Union>c"

163 and xz: "(x, z) \<in> \<Union>c"

164 shows "z = y"

165 proof -

166 from cM have c: "c \<subseteq> M" ..

167 from xy obtain G1 where xy': "(x, y) \<in> G1" and G1: "G1 \<in> c" ..

168 from xz obtain G2 where xz': "(x, z) \<in> G2" and G2: "G2 \<in> c" ..

170 from G1 c have "G1 \<in> M" ..

171 then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"

172 unfolding M_def by blast

174 from G2 c have "G2 \<in> M" ..

175 then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"

176 unfolding M_def by blast

178 txt {* @{text "G\<^sub>1"} is contained in @{text "G\<^sub>2"}

179 or vice versa, since both @{text "G\<^sub>1"} and @{text

180 "G\<^sub>2"} are members of @{text c}. \label{cases2}*}

182 from cM G1 G2 have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") ..

183 then show ?thesis

184 proof

185 assume ?case1

186 with xy' G2_rep have "(x, y) \<in> graph H2 h2" by blast

187 then have "y = h2 x" ..

188 also

189 from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:)

190 then have "z = h2 x" ..

191 finally show ?thesis .

192 next

193 assume ?case2

194 with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast

195 then have "z = h1 x" ..

196 also

197 from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:)

198 then have "y = h1 x" ..

199 finally show ?thesis ..

200 qed

201 qed

203 text {*

204 \medskip The limit function @{text h} is linear. Every element

205 @{text x} in the domain of @{text h} is in the domain of a function

206 @{text h'} in the chain of norm preserving extensions. Furthermore,

207 @{text h} is an extension of @{text h'} so the function values of

208 @{text x} are identical for @{text h'} and @{text h}. Finally, the

209 function @{text h'} is linear by construction of @{text M}.

210 *}

212 lemma sup_lf:

213 assumes M: "M = norm_pres_extensions E p F f"

214 and cM: "c \<in> chains M"

215 and u: "graph H h = \<Union>c"

216 shows "linearform H h"

217 proof

218 fix x y assume x: "x \<in> H" and y: "y \<in> H"

219 with M cM u obtain H' h' where

220 x': "x \<in> H'" and y': "y \<in> H'"

221 and b: "graph H' h' \<subseteq> graph H h"

222 and linearform: "linearform H' h'"

223 and subspace: "H' \<unlhd> E"

224 by (rule some_H'h'2 [elim_format]) blast

226 show "h (x + y) = h x + h y"

227 proof -

228 from linearform x' y' have "h' (x + y) = h' x + h' y"

229 by (rule linearform.add)

230 also from b x' have "h' x = h x" ..

231 also from b y' have "h' y = h y" ..

232 also from subspace x' y' have "x + y \<in> H'"

233 by (rule subspace.add_closed)

234 with b have "h' (x + y) = h (x + y)" ..

235 finally show ?thesis .

236 qed

237 next

238 fix x a assume x: "x \<in> H"

239 with M cM u obtain H' h' where

240 x': "x \<in> H'"

241 and b: "graph H' h' \<subseteq> graph H h"

242 and linearform: "linearform H' h'"

243 and subspace: "H' \<unlhd> E"

244 by (rule some_H'h' [elim_format]) blast

246 show "h (a \<cdot> x) = a * h x"

247 proof -

248 from linearform x' have "h' (a \<cdot> x) = a * h' x"

249 by (rule linearform.mult)

250 also from b x' have "h' x = h x" ..

251 also from subspace x' have "a \<cdot> x \<in> H'"

252 by (rule subspace.mult_closed)

253 with b have "h' (a \<cdot> x) = h (a \<cdot> x)" ..

254 finally show ?thesis .

255 qed

256 qed

258 text {*

259 \medskip The limit of a non-empty chain of norm preserving

260 extensions of @{text f} is an extension of @{text f}, since every

261 element of the chain is an extension of @{text f} and the supremum

262 is an extension for every element of the chain.

263 *}

265 lemma sup_ext:

266 assumes graph: "graph H h = \<Union>c"

267 and M: "M = norm_pres_extensions E p F f"

268 and cM: "c \<in> chains M"

269 and ex: "\<exists>x. x \<in> c"

270 shows "graph F f \<subseteq> graph H h"

271 proof -

272 from ex obtain x where xc: "x \<in> c" ..

273 from cM have "c \<subseteq> M" ..

274 with xc have "x \<in> M" ..

275 with M have "x \<in> norm_pres_extensions E p F f"

276 by (simp only:)

277 then obtain G g where "x = graph G g" and "graph F f \<subseteq> graph G g" ..

278 then have "graph F f \<subseteq> x" by (simp only:)

279 also from xc have "\<dots> \<subseteq> \<Union>c" by blast

280 also from graph have "\<dots> = graph H h" ..

281 finally show ?thesis .

282 qed

284 text {*

285 \medskip The domain @{text H} of the limit function is a superspace

286 of @{text F}, since @{text F} is a subset of @{text H}. The

287 existence of the @{text 0} element in @{text F} and the closure

288 properties follow from the fact that @{text F} is a vector space.

289 *}

291 lemma sup_supF:

292 assumes graph: "graph H h = \<Union>c"

293 and M: "M = norm_pres_extensions E p F f"

294 and cM: "c \<in> chains M"

295 and ex: "\<exists>x. x \<in> c"

296 and FE: "F \<unlhd> E"

297 shows "F \<unlhd> H"

298 proof

299 from FE show "F \<noteq> {}" by (rule subspace.non_empty)

300 from graph M cM ex have "graph F f \<subseteq> graph H h" by (rule sup_ext)

301 then show "F \<subseteq> H" ..

302 fix x y assume "x \<in> F" and "y \<in> F"

303 with FE show "x + y \<in> F" by (rule subspace.add_closed)

304 next

305 fix x a assume "x \<in> F"

306 with FE show "a \<cdot> x \<in> F" by (rule subspace.mult_closed)

307 qed

309 text {*

310 \medskip The domain @{text H} of the limit function is a subspace of

311 @{text E}.

312 *}

314 lemma sup_subE:

315 assumes graph: "graph H h = \<Union>c"

316 and M: "M = norm_pres_extensions E p F f"

317 and cM: "c \<in> chains M"

318 and ex: "\<exists>x. x \<in> c"

319 and FE: "F \<unlhd> E"

320 and E: "vectorspace E"

321 shows "H \<unlhd> E"

322 proof

323 show "H \<noteq> {}"

324 proof -

325 from FE E have "0 \<in> F" by (rule subspace.zero)

326 also from graph M cM ex FE have "F \<unlhd> H" by (rule sup_supF)

327 then have "F \<subseteq> H" ..

328 finally show ?thesis by blast

329 qed

330 show "H \<subseteq> E"

331 proof

332 fix x assume "x \<in> H"

333 with M cM graph

334 obtain H' where x: "x \<in> H'" and H'E: "H' \<unlhd> E"

335 by (rule some_H'h' [elim_format]) blast

336 from H'E have "H' \<subseteq> E" ..

337 with x show "x \<in> E" ..

338 qed

339 fix x y assume x: "x \<in> H" and y: "y \<in> H"

340 show "x + y \<in> H"

341 proof -

342 from M cM graph x y obtain H' h' where

343 x': "x \<in> H'" and y': "y \<in> H'" and H'E: "H' \<unlhd> E"

344 and graphs: "graph H' h' \<subseteq> graph H h"

345 by (rule some_H'h'2 [elim_format]) blast

346 from H'E x' y' have "x + y \<in> H'"

347 by (rule subspace.add_closed)

348 also from graphs have "H' \<subseteq> H" ..

349 finally show ?thesis .

350 qed

351 next

352 fix x a assume x: "x \<in> H"

353 show "a \<cdot> x \<in> H"

354 proof -

355 from M cM graph x

356 obtain H' h' where x': "x \<in> H'" and H'E: "H' \<unlhd> E"

357 and graphs: "graph H' h' \<subseteq> graph H h"

358 by (rule some_H'h' [elim_format]) blast

359 from H'E x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed)

360 also from graphs have "H' \<subseteq> H" ..

361 finally show ?thesis .

362 qed

363 qed

365 text {*

366 \medskip The limit function is bounded by the norm @{text p} as

367 well, since all elements in the chain are bounded by @{text p}.

368 *}

370 lemma sup_norm_pres:

371 assumes graph: "graph H h = \<Union>c"

372 and M: "M = norm_pres_extensions E p F f"

373 and cM: "c \<in> chains M"

374 shows "\<forall>x \<in> H. h x \<le> p x"

375 proof

376 fix x assume "x \<in> H"

377 with M cM graph obtain H' h' where x': "x \<in> H'"

378 and graphs: "graph H' h' \<subseteq> graph H h"

379 and a: "\<forall>x \<in> H'. h' x \<le> p x"

380 by (rule some_H'h' [elim_format]) blast

381 from graphs x' have [symmetric]: "h' x = h x" ..

382 also from a x' have "h' x \<le> p x " ..

383 finally show "h x \<le> p x" .

384 qed

386 text {*

387 \medskip The following lemma is a property of linear forms on real

388 vector spaces. It will be used for the lemma @{text abs_Hahn_Banach}

389 (see page \pageref{abs-Hahn_Banach}). \label{abs-ineq-iff} For real

390 vector spaces the following inequations are equivalent:

391 \begin{center}

392 \begin{tabular}{lll}

393 @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &

394 @{text "\<forall>x \<in> H. h x \<le> p x"} \\

395 \end{tabular}

396 \end{center}

397 *}

399 lemma abs_ineq_iff:

400 assumes "subspace H E" and "vectorspace E" and "seminorm E p"

401 and "linearform H h"

402 shows "(\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)" (is "?L = ?R")

403 proof

404 interpret subspace H E by fact

405 interpret vectorspace E by fact

406 interpret seminorm E p by fact

407 interpret linearform H h by fact

408 have H: "vectorspace H" using `vectorspace E` ..

409 {

410 assume l: ?L

411 show ?R

412 proof

413 fix x assume x: "x \<in> H"

414 have "h x \<le> \<bar>h x\<bar>" by arith

415 also from l x have "\<dots> \<le> p x" ..

416 finally show "h x \<le> p x" .

417 qed

418 next

419 assume r: ?R

420 show ?L

421 proof

422 fix x assume x: "x \<in> H"

423 show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a"

424 by arith

425 from `linearform H h` and H x

426 have "- h x = h (- x)" by (rule linearform.neg [symmetric])

427 also

428 from H x have "- x \<in> H" by (rule vectorspace.neg_closed)

429 with r have "h (- x) \<le> p (- x)" ..

430 also have "\<dots> = p x"

431 using `seminorm E p` `vectorspace E`

432 proof (rule seminorm.minus)

433 from x show "x \<in> E" ..

434 qed

435 finally have "- h x \<le> p x" .

436 then show "- p x \<le> h x" by simp

437 from r x show "h x \<le> p x" ..

438 qed

439 }

440 qed

442 end