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src/HOL/Hahn_Banach/Hahn_Banach.thy

author | hoelzl |

Thu Sep 02 10:14:32 2010 +0200 (2010-09-02) | |

changeset 39072 | 1030b1a166ef |

parent 32960 | 69916a850301 |

child 44190 | fe5504984937 |

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

Add lessThan_Suc_eq_insert_0

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

2 Author: Gertrud Bauer, TU Munich

3 *)

5 header {* The Hahn-Banach Theorem *}

7 theory Hahn_Banach

8 imports Hahn_Banach_Lemmas

9 begin

11 text {*

12 We present the proof of two different versions of the Hahn-Banach

13 Theorem, closely following \cite[\S36]{Heuser:1986}.

14 *}

16 subsection {* The Hahn-Banach Theorem for vector spaces *}

18 text {*

19 \textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real

20 vector space @{text E}, let @{text p} be a semi-norm on @{text E},

21 and @{text f} be a linear form defined on @{text F} such that @{text

22 f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in> F. f x \<le> p x"}. Then

23 @{text f} can be extended to a linear form @{text h} on @{text E}

24 such that @{text h} is norm-preserving, i.e. @{text h} is also

25 bounded by @{text p}.

27 \bigskip

28 \textbf{Proof Sketch.}

29 \begin{enumerate}

31 \item Define @{text M} as the set of norm-preserving extensions of

32 @{text f} to subspaces of @{text E}. The linear forms in @{text M}

33 are ordered by domain extension.

35 \item We show that every non-empty chain in @{text M} has an upper

36 bound in @{text M}.

38 \item With Zorn's Lemma we conclude that there is a maximal function

39 @{text g} in @{text M}.

41 \item The domain @{text H} of @{text g} is the whole space @{text

42 E}, as shown by classical contradiction:

44 \begin{itemize}

46 \item Assuming @{text g} is not defined on whole @{text E}, it can

47 still be extended in a norm-preserving way to a super-space @{text

48 H'} of @{text H}.

50 \item Thus @{text g} can not be maximal. Contradiction!

52 \end{itemize}

53 \end{enumerate}

54 *}

56 theorem Hahn_Banach:

57 assumes E: "vectorspace E" and "subspace F E"

58 and "seminorm E p" and "linearform F f"

59 assumes fp: "\<forall>x \<in> F. f x \<le> p x"

60 shows "\<exists>h. linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)"

61 -- {* Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E}, *}

62 -- {* and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p}, *}

63 -- {* then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \skp *}

64 proof -

65 interpret vectorspace E by fact

66 interpret subspace F E by fact

67 interpret seminorm E p by fact

68 interpret linearform F f by fact

69 def M \<equiv> "norm_pres_extensions E p F f"

70 then have M: "M = \<dots>" by (simp only:)

71 from E have F: "vectorspace F" ..

72 note FE = `F \<unlhd> E`

73 {

74 fix c assume cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c"

75 have "\<Union>c \<in> M"

76 -- {* Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}: *}

77 -- {* @{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}. *}

78 unfolding M_def

79 proof (rule norm_pres_extensionI)

80 let ?H = "domain (\<Union>c)"

81 let ?h = "funct (\<Union>c)"

83 have a: "graph ?H ?h = \<Union>c"

84 proof (rule graph_domain_funct)

85 fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"

86 with M_def cM show "z = y" by (rule sup_definite)

87 qed

88 moreover from M cM a have "linearform ?H ?h"

89 by (rule sup_lf)

90 moreover from a M cM ex FE E have "?H \<unlhd> E"

91 by (rule sup_subE)

92 moreover from a M cM ex FE have "F \<unlhd> ?H"

93 by (rule sup_supF)

94 moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"

95 by (rule sup_ext)

96 moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"

97 by (rule sup_norm_pres)

98 ultimately show "\<exists>H h. \<Union>c = graph H h

99 \<and> linearform H h

100 \<and> H \<unlhd> E

101 \<and> F \<unlhd> H

102 \<and> graph F f \<subseteq> graph H h

103 \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast

104 qed

105 }

106 then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"

107 -- {* With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp *}

108 proof (rule Zorn's_Lemma)

109 -- {* We show that @{text M} is non-empty: *}

110 show "graph F f \<in> M"

111 unfolding M_def

112 proof (rule norm_pres_extensionI2)

113 show "linearform F f" by fact

114 show "F \<unlhd> E" by fact

115 from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)

116 show "graph F f \<subseteq> graph F f" ..

117 show "\<forall>x\<in>F. f x \<le> p x" by fact

118 qed

119 qed

120 then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"

121 by blast

122 from gM obtain H h where

123 g_rep: "g = graph H h"

124 and linearform: "linearform H h"

125 and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"

126 and graphs: "graph F f \<subseteq> graph H h"

127 and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..

128 -- {* @{text g} is a norm-preserving extension of @{text f}, in other words: *}

129 -- {* @{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E}, *}

130 -- {* and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp *}

131 from HE E have H: "vectorspace H"

132 by (rule subspace.vectorspace)

134 have HE_eq: "H = E"

135 -- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}

136 proof (rule classical)

137 assume neq: "H \<noteq> E"

138 -- {* Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended *}

139 -- {* in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp *}

140 have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"

141 proof -

142 from HE have "H \<subseteq> E" ..

143 with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast

144 obtain x': "x' \<noteq> 0"

145 proof

146 show "x' \<noteq> 0"

147 proof

148 assume "x' = 0"

149 with H have "x' \<in> H" by (simp only: vectorspace.zero)

150 with `x' \<notin> H` show False by contradiction

151 qed

152 qed

154 def H' \<equiv> "H + lin x'"

155 -- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}

156 have HH': "H \<unlhd> H'"

157 proof (unfold H'_def)

158 from x'E have "vectorspace (lin x')" ..

159 with H show "H \<unlhd> H + lin x'" ..

160 qed

162 obtain xi where

163 xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi

164 \<and> xi \<le> p (y + x') - h y"

165 -- {* Pick a real number @{text \<xi>} that fulfills certain inequations; this will *}

166 -- {* be used to establish that @{text h'} is a norm-preserving extension of @{text h}.

167 \label{ex-xi-use}\skp *}

168 proof -

169 from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi

170 \<and> xi \<le> p (y + x') - h y"

171 proof (rule ex_xi)

172 fix u v assume u: "u \<in> H" and v: "v \<in> H"

173 with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto

174 from H u v linearform have "h v - h u = h (v - u)"

175 by (simp add: linearform.diff)

176 also from hp and H u v have "\<dots> \<le> p (v - u)"

177 by (simp only: vectorspace.diff_closed)

178 also from x'E uE vE have "v - u = x' + - x' + v + - u"

179 by (simp add: diff_eq1)

180 also from x'E uE vE have "\<dots> = v + x' + - (u + x')"

181 by (simp add: add_ac)

182 also from x'E uE vE have "\<dots> = (v + x') - (u + x')"

183 by (simp add: diff_eq1)

184 also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"

185 by (simp add: diff_subadditive)

186 finally have "h v - h u \<le> p (v + x') + p (u + x')" .

187 then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp

188 qed

189 then show thesis by (blast intro: that)

190 qed

192 def h' \<equiv> "\<lambda>x. let (y, a) =

193 SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"

194 -- {* Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp *}

196 have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"

197 -- {* @{text h'} is an extension of @{text h} \dots \skp *}

198 proof

199 show "g \<subseteq> graph H' h'"

200 proof -

201 have "graph H h \<subseteq> graph H' h'"

202 proof (rule graph_extI)

203 fix t assume t: "t \<in> H"

204 from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"

205 using `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` by (rule decomp_H'_H)

206 with h'_def show "h t = h' t" by (simp add: Let_def)

207 next

208 from HH' show "H \<subseteq> H'" ..

209 qed

210 with g_rep show ?thesis by (simp only:)

211 qed

213 show "g \<noteq> graph H' h'"

214 proof -

215 have "graph H h \<noteq> graph H' h'"

216 proof

217 assume eq: "graph H h = graph H' h'"

218 have "x' \<in> H'"

219 unfolding H'_def

220 proof

221 from H show "0 \<in> H" by (rule vectorspace.zero)

222 from x'E show "x' \<in> lin x'" by (rule x_lin_x)

223 from x'E show "x' = 0 + x'" by simp

224 qed

225 then have "(x', h' x') \<in> graph H' h'" ..

226 with eq have "(x', h' x') \<in> graph H h" by (simp only:)

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

228 with `x' \<notin> H` show False by contradiction

229 qed

230 with g_rep show ?thesis by simp

231 qed

232 qed

233 moreover have "graph H' h' \<in> M"

234 -- {* and @{text h'} is norm-preserving. \skp *}

235 proof (unfold M_def)

236 show "graph H' h' \<in> norm_pres_extensions E p F f"

237 proof (rule norm_pres_extensionI2)

238 show "linearform H' h'"

239 using h'_def H'_def HE linearform `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E

240 by (rule h'_lf)

241 show "H' \<unlhd> E"

242 unfolding H'_def

243 proof

244 show "H \<unlhd> E" by fact

245 show "vectorspace E" by fact

246 from x'E show "lin x' \<unlhd> E" ..

247 qed

248 from H `F \<unlhd> H` HH' show FH': "F \<unlhd> H'"

249 by (rule vectorspace.subspace_trans)

250 show "graph F f \<subseteq> graph H' h'"

251 proof (rule graph_extI)

252 fix x assume x: "x \<in> F"

253 with graphs have "f x = h x" ..

254 also have "\<dots> = h x + 0 * xi" by simp

255 also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"

256 by (simp add: Let_def)

257 also have "(x, 0) =

258 (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"

259 using E HE

260 proof (rule decomp_H'_H [symmetric])

261 from FH x show "x \<in> H" ..

262 from x' show "x' \<noteq> 0" .

263 show "x' \<notin> H" by fact

264 show "x' \<in> E" by fact

265 qed

266 also have

267 "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)

268 in h y + a * xi) = h' x" by (simp only: h'_def)

269 finally show "f x = h' x" .

270 next

271 from FH' show "F \<subseteq> H'" ..

272 qed

273 show "\<forall>x \<in> H'. h' x \<le> p x"

274 using h'_def H'_def `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E HE

275 `seminorm E p` linearform and hp xi

276 by (rule h'_norm_pres)

277 qed

278 qed

279 ultimately show ?thesis ..

280 qed

281 then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp

282 -- {* So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp *}

283 with gx show "H = E" by contradiction

284 qed

286 from HE_eq and linearform have "linearform E h"

287 by (simp only:)

288 moreover have "\<forall>x \<in> F. h x = f x"

289 proof

290 fix x assume "x \<in> F"

291 with graphs have "f x = h x" ..

292 then show "h x = f x" ..

293 qed

294 moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"

295 by (simp only:)

296 ultimately show ?thesis by blast

297 qed

300 subsection {* Alternative formulation *}

302 text {*

303 The following alternative formulation of the Hahn-Banach

304 Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear

305 form @{text f} and a seminorm @{text p} the following inequations

306 are equivalent:\footnote{This was shown in lemma @{thm [source]

307 abs_ineq_iff} (see page \pageref{abs-ineq-iff}).}

308 \begin{center}

309 \begin{tabular}{lll}

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

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

312 \end{tabular}

313 \end{center}

314 *}

316 theorem abs_Hahn_Banach:

317 assumes E: "vectorspace E" and FE: "subspace F E"

318 and lf: "linearform F f" and sn: "seminorm E p"

319 assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

320 shows "\<exists>g. linearform E g

321 \<and> (\<forall>x \<in> F. g x = f x)

322 \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"

323 proof -

324 interpret vectorspace E by fact

325 interpret subspace F E by fact

326 interpret linearform F f by fact

327 interpret seminorm E p by fact

328 have "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x) \<and> (\<forall>x \<in> E. g x \<le> p x)"

329 using E FE sn lf

330 proof (rule Hahn_Banach)

331 show "\<forall>x \<in> F. f x \<le> p x"

332 using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])

333 qed

334 then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"

335 and **: "\<forall>x \<in> E. g x \<le> p x" by blast

336 have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

337 using _ E sn lg **

338 proof (rule abs_ineq_iff [THEN iffD2])

339 show "E \<unlhd> E" ..

340 qed

341 with lg * show ?thesis by blast

342 qed

345 subsection {* The Hahn-Banach Theorem for normed spaces *}

347 text {*

348 Every continuous linear form @{text f} on a subspace @{text F} of a

349 norm space @{text E}, can be extended to a continuous linear form

350 @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}.

351 *}

353 theorem norm_Hahn_Banach:

354 fixes V and norm ("\<parallel>_\<parallel>")

355 fixes B defines "\<And>V f. B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"

356 fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)

357 defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"

358 assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"

359 and linearform: "linearform F f" and "continuous F norm f"

360 shows "\<exists>g. linearform E g

361 \<and> continuous E norm g

362 \<and> (\<forall>x \<in> F. g x = f x)

363 \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

364 proof -

365 interpret normed_vectorspace E norm by fact

366 interpret normed_vectorspace_with_fn_norm E norm B fn_norm

367 by (auto simp: B_def fn_norm_def) intro_locales

368 interpret subspace F E by fact

369 interpret linearform F f by fact

370 interpret continuous F norm f by fact

371 have E: "vectorspace E" by intro_locales

372 have F: "vectorspace F" by rule intro_locales

373 have F_norm: "normed_vectorspace F norm"

374 using FE E_norm by (rule subspace_normed_vs)

375 have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"

376 by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero

377 [OF normed_vectorspace_with_fn_norm.intro,

378 OF F_norm `continuous F norm f` , folded B_def fn_norm_def])

379 txt {* We define a function @{text p} on @{text E} as follows:

380 @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"} *}

381 def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

383 txt {* @{text p} is a seminorm on @{text E}: *}

384 have q: "seminorm E p"

385 proof

386 fix x y a assume x: "x \<in> E" and y: "y \<in> E"

388 txt {* @{text p} is positive definite: *}

389 have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

390 moreover from x have "0 \<le> \<parallel>x\<parallel>" ..

391 ultimately show "0 \<le> p x"

392 by (simp add: p_def zero_le_mult_iff)

394 txt {* @{text p} is absolutely homogenous: *}

396 show "p (a \<cdot> x) = \<bar>a\<bar> * p x"

397 proof -

398 have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)

399 also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)

400 also have "\<parallel>f\<parallel>\<hyphen>F * (\<bar>a\<bar> * \<parallel>x\<parallel>) = \<bar>a\<bar> * (\<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>)" by simp

401 also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)

402 finally show ?thesis .

403 qed

405 txt {* Furthermore, @{text p} is subadditive: *}

407 show "p (x + y) \<le> p x + p y"

408 proof -

409 have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)

410 also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

411 from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..

412 with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"

413 by (simp add: mult_left_mono)

414 also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: right_distrib)

415 also have "\<dots> = p x + p y" by (simp only: p_def)

416 finally show ?thesis .

417 qed

418 qed

420 txt {* @{text f} is bounded by @{text p}. *}

422 have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

423 proof

424 fix x assume "x \<in> F"

425 with `continuous F norm f` and linearform

426 show "\<bar>f x\<bar> \<le> p x"

427 unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong

428 [OF normed_vectorspace_with_fn_norm.intro,

429 OF F_norm, folded B_def fn_norm_def])

430 qed

432 txt {* Using the fact that @{text p} is a seminorm and @{text f} is bounded

433 by @{text p} we can apply the Hahn-Banach Theorem for real vector

434 spaces. So @{text f} can be extended in a norm-preserving way to

435 some function @{text g} on the whole vector space @{text E}. *}

437 with E FE linearform q obtain g where

438 linearformE: "linearform E g"

439 and a: "\<forall>x \<in> F. g x = f x"

440 and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

441 by (rule abs_Hahn_Banach [elim_format]) iprover

443 txt {* We furthermore have to show that @{text g} is also continuous: *}

445 have g_cont: "continuous E norm g" using linearformE

446 proof

447 fix x assume "x \<in> E"

448 with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

449 by (simp only: p_def)

450 qed

452 txt {* To complete the proof, we show that @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. *}

454 have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

455 proof (rule order_antisym)

456 txt {*

457 First we show @{text "\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>"}. The function norm @{text

458 "\<parallel>g\<parallel>"} is defined as the smallest @{text "c \<in> \<real>"} such that

459 \begin{center}

460 \begin{tabular}{l}

461 @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}

462 \end{tabular}

463 \end{center}

464 \noindent Furthermore holds

465 \begin{center}

466 \begin{tabular}{l}

467 @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}

468 \end{tabular}

469 \end{center}

470 *}

472 have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

473 proof

474 fix x assume "x \<in> E"

475 with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

476 by (simp only: p_def)

477 qed

478 from g_cont this ge_zero

479 show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"

480 by (rule fn_norm_least [of g, folded B_def fn_norm_def])

482 txt {* The other direction is achieved by a similar argument. *}

484 show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"

485 proof (rule normed_vectorspace_with_fn_norm.fn_norm_least

486 [OF normed_vectorspace_with_fn_norm.intro,

487 OF F_norm, folded B_def fn_norm_def])

488 show "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

489 proof

490 fix x assume x: "x \<in> F"

491 from a x have "g x = f x" ..

492 then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)

493 also from g_cont

494 have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

495 proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])

496 from FE x show "x \<in> E" ..

497 qed

498 finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .

499 qed

500 show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"

501 using g_cont

502 by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])

503 show "continuous F norm f" by fact

504 qed

505 qed

506 with linearformE a g_cont show ?thesis by blast

507 qed

509 end