src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy
 author ballarin Tue Jul 15 16:50:09 2008 +0200 (2008-07-15) changeset 27611 2c01c0bdb385 parent 23378 1d138d6bb461 child 27612 d3eb431db035 permissions -rw-r--r--
Removed uses of context element includes.
     1 (*  Title:      HOL/Real/HahnBanach/HahnBanachExtLemmas.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6 header {* Extending non-maximal functions *}

     7

     8 theory HahnBanachExtLemmas imports FunctionNorm begin

     9

    10 text {*

    11   In this section the following context is presumed.  Let @{text E} be

    12   a real vector space with a seminorm @{text q} on @{text E}. @{text

    13   F} is a subspace of @{text E} and @{text f} a linear function on

    14   @{text F}. We consider a subspace @{text H} of @{text E} that is a

    15   superspace of @{text F} and a linear form @{text h} on @{text

    16   H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is

    17   an element in @{text "E - H"}.  @{text H} is extended to the direct

    18   sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"}

    19   the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is

    20   unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y +

    21   a \<cdot> \<xi>"} for a certain @{text \<xi>}.

    22

    23   Subsequently we show some properties of this extension @{text h'} of

    24   @{text h}.

    25

    26   \medskip This lemma will be used to show the existence of a linear

    27   extension of @{text f} (see page \pageref{ex-xi-use}). It is a

    28   consequence of the completeness of @{text \<real>}. To show

    29   \begin{center}

    30   \begin{tabular}{l}

    31   @{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}

    32   \end{tabular}

    33   \end{center}

    34   \noindent it suffices to show that

    35   \begin{center}

    36   \begin{tabular}{l}

    37   @{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"}

    38   \end{tabular}

    39   \end{center}

    40 *}

    41

    42 lemma ex_xi:

    43   assumes "vectorspace F"

    44   assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"

    45   shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"

    46 proof -

    47   interpret vectorspace [F] by fact

    48   txt {* From the completeness of the reals follows:

    49     The set @{text "S = {a u. u \<in> F}"} has a supremum, if it is

    50     non-empty and has an upper bound. *}

    51

    52   let ?S = "{a u | u. u \<in> F}"

    53   have "\<exists>xi. lub ?S xi"

    54   proof (rule real_complete)

    55     have "a 0 \<in> ?S" by blast

    56     then show "\<exists>X. X \<in> ?S" ..

    57     have "\<forall>y \<in> ?S. y \<le> b 0"

    58     proof

    59       fix y assume y: "y \<in> ?S"

    60       then obtain u where u: "u \<in> F" and y: "y = a u" by blast

    61       from u and zero have "a u \<le> b 0" by (rule r)

    62       with y show "y \<le> b 0" by (simp only:)

    63     qed

    64     then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..

    65   qed

    66   then obtain xi where xi: "lub ?S xi" ..

    67   {

    68     fix y assume "y \<in> F"

    69     then have "a y \<in> ?S" by blast

    70     with xi have "a y \<le> xi" by (rule lub.upper)

    71   } moreover {

    72     fix y assume y: "y \<in> F"

    73     from xi have "xi \<le> b y"

    74     proof (rule lub.least)

    75       fix au assume "au \<in> ?S"

    76       then obtain u where u: "u \<in> F" and au: "au = a u" by blast

    77       from u y have "a u \<le> b y" by (rule r)

    78       with au show "au \<le> b y" by (simp only:)

    79     qed

    80   } ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast

    81 qed

    82

    83 text {*

    84   \medskip The function @{text h'} is defined as a @{text "h' x = h y

    85   + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of

    86   @{text h} to @{text H'}.

    87 *}

    88

    89 lemma h'_lf:

    90   assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =

    91       SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"

    92     and H'_def: "H' \<equiv> H + lin x0"

    93     and HE: "H \<unlhd> E"

    94   assumes "linearform H h"

    95   assumes x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"

    96   assumes E: "vectorspace E"

    97   shows "linearform H' h'"

    98 proof -

    99   interpret linearform [H h] by fact

   100   interpret vectorspace [E] by fact

   101   show ?thesis proof

   102     note E = vectorspace E

   103     have H': "vectorspace H'"

   104     proof (unfold H'_def)

   105       from x0 \<in> E

   106       have "lin x0 \<unlhd> E" ..

   107       with HE show "vectorspace (H + lin x0)" using E ..

   108     qed

   109     {

   110       fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"

   111       show "h' (x1 + x2) = h' x1 + h' x2"

   112       proof -

   113 	from H' x1 x2 have "x1 + x2 \<in> H'"

   114           by (rule vectorspace.add_closed)

   115 	with x1 x2 obtain y y1 y2 a a1 a2 where

   116           x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"

   117           and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

   118           and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"

   119           by (unfold H'_def sum_def lin_def) blast

   120

   121 	have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0

   122 	proof (rule decomp_H') txt_raw {* \label{decomp-H-use} *}

   123           from HE y1 y2 show "y1 + y2 \<in> H"

   124             by (rule subspace.add_closed)

   125           from x0 and HE y y1 y2

   126           have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E"  "y2 \<in> E" by auto

   127           with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"

   128             by (simp add: add_ac add_mult_distrib2)

   129           also note x1x2

   130           finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .

   131 	qed

   132

   133 	from h'_def x1x2 E HE y x0

   134 	have "h' (x1 + x2) = h y + a * xi"

   135           by (rule h'_definite)

   136 	also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"

   137           by (simp only: ya)

   138 	also from y1 y2 have "h (y1 + y2) = h y1 + h y2"

   139           by simp

   140 	also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"

   141           by (simp add: left_distrib)

   142 	also from h'_def x1_rep E HE y1 x0

   143 	have "h y1 + a1 * xi = h' x1"

   144           by (rule h'_definite [symmetric])

   145 	also from h'_def x2_rep E HE y2 x0

   146 	have "h y2 + a2 * xi = h' x2"

   147           by (rule h'_definite [symmetric])

   148 	finally show ?thesis .

   149       qed

   150     next

   151       fix x1 c assume x1: "x1 \<in> H'"

   152       show "h' (c \<cdot> x1) = c * (h' x1)"

   153       proof -

   154 	from H' x1 have ax1: "c \<cdot> x1 \<in> H'"

   155           by (rule vectorspace.mult_closed)

   156 	with x1 obtain y a y1 a1 where

   157           cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"

   158           and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

   159           by (unfold H'_def sum_def lin_def) blast

   160

   161 	have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0

   162 	proof (rule decomp_H')

   163           from HE y1 show "c \<cdot> y1 \<in> H"

   164             by (rule subspace.mult_closed)

   165           from x0 and HE y y1

   166           have "x0 \<in> E"  "y \<in> E"  "y1 \<in> E" by auto

   167           with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"

   168             by (simp add: mult_assoc add_mult_distrib1)

   169           also note cx1_rep

   170           finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .

   171 	qed

   172

   173 	from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"

   174           by (rule h'_definite)

   175 	also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"

   176           by (simp only: ya)

   177 	also from y1 have "h (c \<cdot> y1) = c * h y1"

   178           by simp

   179 	also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"

   180           by (simp only: right_distrib)

   181 	also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"

   182           by (rule h'_definite [symmetric])

   183 	finally show ?thesis .

   184       qed

   185     }

   186   qed

   187 qed

   188

   189 text {* \medskip The linear extension @{text h'} of @{text h}

   190   is bounded by the seminorm @{text p}. *}

   191

   192 lemma h'_norm_pres:

   193   assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =

   194       SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"

   195     and H'_def: "H' \<equiv> H + lin x0"

   196     and x0: "x0 \<notin> H"  "x0 \<in> E"  "x0 \<noteq> 0"

   197   assumes E: "vectorspace E" and HE: "subspace H E"

   198     and "seminorm E p" and "linearform H h"

   199   assumes a: "\<forall>y \<in> H. h y \<le> p y"

   200     and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"

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

   202 proof -

   203   interpret vectorspace [E] by fact

   204   interpret subspace [H E] by fact

   205   interpret seminorm [E p] by fact

   206   interpret linearform [H h] by fact

   207   show ?thesis proof

   208     fix x assume x': "x \<in> H'"

   209     show "h' x \<le> p x"

   210     proof -

   211       from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"

   212 	and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto

   213       from x' obtain y a where

   214         x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"

   215 	by (unfold H'_def sum_def lin_def) blast

   216       from y have y': "y \<in> E" ..

   217       from y have ay: "inverse a \<cdot> y \<in> H" by simp

   218

   219       from h'_def x_rep E HE y x0 have "h' x = h y + a * xi"

   220 	by (rule h'_definite)

   221       also have "\<dots> \<le> p (y + a \<cdot> x0)"

   222       proof (rule linorder_cases)

   223 	assume z: "a = 0"

   224 	then have "h y + a * xi = h y" by simp

   225 	also from a y have "\<dots> \<le> p y" ..

   226 	also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp

   227 	finally show ?thesis .

   228       next

   229 	txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}

   230           with @{text ya} taken as @{text "y / a"}: *}

   231 	assume lz: "a < 0" hence nz: "a \<noteq> 0" by simp

   232 	from a1 ay

   233 	have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..

   234 	with lz have "a * xi \<le>

   235           a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

   236           by (simp add: mult_left_mono_neg order_less_imp_le)

   237

   238 	also have "\<dots> =

   239           - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"

   240 	  by (simp add: right_diff_distrib)

   241 	also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =

   242           p (a \<cdot> (inverse a \<cdot> y + x0))"

   243           by (simp add: abs_homogenous)

   244 	also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

   245           by (simp add: add_mult_distrib1 mult_assoc [symmetric])

   246 	also from nz y have "a * (h (inverse a \<cdot> y)) =  h y"

   247           by simp

   248 	finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

   249 	then show ?thesis by simp

   250       next

   251 	txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}

   252           with @{text ya} taken as @{text "y / a"}: *}

   253 	assume gz: "0 < a" hence nz: "a \<noteq> 0" by simp

   254 	from a2 ay

   255 	have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..

   256 	with gz have "a * xi \<le>

   257           a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

   258           by simp

   259 	also have "... = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"

   260 	  by (simp add: right_diff_distrib)

   261 	also from gz x0 y'

   262 	have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"

   263           by (simp add: abs_homogenous)

   264 	also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

   265           by (simp add: add_mult_distrib1 mult_assoc [symmetric])

   266 	also from nz y have "a * h (inverse a \<cdot> y) = h y"

   267           by simp

   268 	finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

   269 	then show ?thesis by simp

   270       qed

   271       also from x_rep have "\<dots> = p x" by (simp only:)

   272       finally show ?thesis .

   273     qed

   274   qed

   275 qed

   276

   277 end