Merged.
1 (* Title: HOL/Real/HahnBanach/HahnBanach.thy
2 Author: Gertrud Bauer, TU Munich
5 header {* The Hahn-Banach Theorem *}
8 imports HahnBanachLemmas
12 We present the proof of two different versions of the Hahn-Banach
13 Theorem, closely following \cite[\S36]{Heuser:1986}.
16 subsection {* The Hahn-Banach Theorem for vector spaces *}
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
28 \textbf{Proof Sketch.}
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
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:
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
50 \item Thus @{text g} can not be maximal. Contradiction!
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 *}
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`
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"}. *}
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)
88 moreover from M cM a have "linearform ?H ?h"
90 moreover from a M cM ex FE E have "?H \<unlhd> E"
92 moreover from a M cM ex FE have "F \<unlhd> ?H"
94 moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
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
102 \<and> graph F f \<subseteq> graph H h
103 \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast
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"
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
120 then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
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)
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'"
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"
149 with H have "x' \<in> H" by (simp only: vectorspace.zero)
150 with `x' \<notin> H` show False by contradiction
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'" ..
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 *}
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"
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
189 then show thesis by (blast intro: that)
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 *}
199 show "g \<subseteq> graph H' h'"
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)
208 from HH' show "H \<subseteq> H'" ..
210 with g_rep show ?thesis by (simp only:)
213 show "g \<noteq> graph H' h'"
215 have "graph H h \<noteq> graph H' h'"
217 assume eq: "graph H h = graph H' h'"
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
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
230 with g_rep show ?thesis by simp
233 moreover have "graph H' h' \<in> M"
234 -- {* and @{text h'} is norm-preserving. \skp *}
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
244 show "H \<unlhd> E" by fact
245 show "vectorspace E" by fact
246 from x'E show "lin x' \<unlhd> E" ..
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)
258 (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"
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
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" .
271 from FH' show "F \<subseteq> H'" ..
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)
279 ultimately show ?thesis ..
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
286 from HE_eq and linearform have "linearform E h"
288 moreover have "\<forall>x \<in> F. h x = f x"
290 fix x assume "x \<in> F"
291 with graphs have "f x = h x" ..
292 then show "h x = f x" ..
294 moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
296 ultimately show ?thesis by blast
300 subsection {* Alternative formulation *}
303 The following alternative formulation of the Hahn-Banach
304 Theorem\label{abs-HahnBanach} 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}).}
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"} \\
316 theorem abs_HahnBanach:
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)"
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)"
330 proof (rule HahnBanach)
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])
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"
338 proof (rule abs_ineq_iff [THEN iffD2])
339 show "E \<unlhd> E" ..
341 with lg * show ?thesis by blast
345 subsection {* The Hahn-Banach Theorem for normed spaces *}
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>"}.
353 theorem norm_HahnBanach:
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"
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"
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"
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 .
405 txt {* Furthermore, @{text p} is subadditive: *}
407 show "p (x + y) \<le> p x + p y"
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 .
420 txt {* @{text f} is bounded by @{text p}. *}
422 have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
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])
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_HahnBanach [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
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)
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)
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
461 @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
464 \noindent Furthermore holds
467 @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
472 have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
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)
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>"
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:)
494 have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
495 proof (rule fn_norm_le_cong [of g, folded B_def fn_norm_def])
496 from FE x show "x \<in> E" ..
498 finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
500 show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
502 by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
503 show "continuous F norm f" by fact
506 with linearformE a g_cont show ?thesis by blast