src/HOL/Hahn_Banach/Function_Norm.thy
 author hoelzl Thu Sep 02 10:14:32 2010 +0200 (2010-09-02) changeset 39072 1030b1a166ef parent 36778 739a9379e29b child 44887 7ca82df6e951 permissions -rw-r--r--
1 (*  Title:      HOL/Hahn_Banach/Function_Norm.thy
2     Author:     Gertrud Bauer, TU Munich
3 *)
5 header {* The norm of a function *}
7 theory Function_Norm
8 imports Normed_Space Function_Order
9 begin
11 subsection {* Continuous linear forms*}
13 text {*
14   A linear form @{text f} on a normed vector space @{text "(V, \<parallel>\<cdot>\<parallel>)"}
15   is \emph{continuous}, iff it is bounded, i.e.
16   \begin{center}
17   @{text "\<exists>c \<in> R. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
18   \end{center}
19   In our application no other functions than linear forms are
20   considered, so we can define continuous linear forms as bounded
21   linear forms:
22 *}
24 locale continuous = var_V + norm_syntax + linearform +
25   assumes bounded: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
27 declare continuous.intro [intro?] continuous_axioms.intro [intro?]
29 lemma continuousI [intro]:
30   fixes norm :: "_ \<Rightarrow> real"  ("\<parallel>_\<parallel>")
31   assumes "linearform V f"
32   assumes r: "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
33   shows "continuous V norm f"
34 proof
35   show "linearform V f" by fact
36   from r have "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" by blast
37   then show "continuous_axioms V norm f" ..
38 qed
41 subsection {* The norm of a linear form *}
43 text {*
44   The least real number @{text c} for which holds
45   \begin{center}
46   @{text "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
47   \end{center}
48   is called the \emph{norm} of @{text f}.
50   For non-trivial vector spaces @{text "V \<noteq> {0}"} the norm can be
51   defined as
52   \begin{center}
53   @{text "\<parallel>f\<parallel> = \<sup>x \<noteq> 0. \<bar>f x\<bar> / \<parallel>x\<parallel>"}
54   \end{center}
56   For the case @{text "V = {0}"} the supremum would be taken from an
57   empty set. Since @{text \<real>} is unbounded, there would be no supremum.
58   To avoid this situation it must be guaranteed that there is an
59   element in this set. This element must be @{text "{} \<ge> 0"} so that
60   @{text fn_norm} has the norm properties. Furthermore it does not
61   have to change the norm in all other cases, so it must be @{text 0},
62   as all other elements are @{text "{} \<ge> 0"}.
64   Thus we define the set @{text B} where the supremum is taken from as
65   follows:
66   \begin{center}
67   @{text "{0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel>. x \<noteq> 0 \<and> x \<in> F}"}
68   \end{center}
70   @{text fn_norm} is equal to the supremum of @{text B}, if the
71   supremum exists (otherwise it is undefined).
72 *}
74 locale fn_norm = norm_syntax +
75   fixes B defines "B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"
76   fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
77   defines "\<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
79 locale normed_vectorspace_with_fn_norm = normed_vectorspace + fn_norm
81 lemma (in fn_norm) B_not_empty [intro]: "0 \<in> B V f"
82   by (simp add: B_def)
84 text {*
85   The following lemma states that every continuous linear form on a
86   normed space @{text "(V, \<parallel>\<cdot>\<parallel>)"} has a function norm.
87 *}
89 lemma (in normed_vectorspace_with_fn_norm) fn_norm_works:
90   assumes "continuous V norm f"
91   shows "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
92 proof -
93   interpret continuous V norm f by fact
94   txt {* The existence of the supremum is shown using the
95     completeness of the reals. Completeness means, that every
96     non-empty bounded set of reals has a supremum. *}
97   have "\<exists>a. lub (B V f) a"
98   proof (rule real_complete)
99     txt {* First we have to show that @{text B} is non-empty: *}
100     have "0 \<in> B V f" ..
101     then show "\<exists>x. x \<in> B V f" ..
103     txt {* Then we have to show that @{text B} is bounded: *}
104     show "\<exists>c. \<forall>y \<in> B V f. y \<le> c"
105     proof -
106       txt {* We know that @{text f} is bounded by some value @{text c}. *}
107       from bounded obtain c where c: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
109       txt {* To prove the thesis, we have to show that there is some
110         @{text b}, such that @{text "y \<le> b"} for all @{text "y \<in>
111         B"}. Due to the definition of @{text B} there are two cases. *}
113       def b \<equiv> "max c 0"
114       have "\<forall>y \<in> B V f. y \<le> b"
115       proof
116         fix y assume y: "y \<in> B V f"
117         show "y \<le> b"
118         proof cases
119           assume "y = 0"
120           then show ?thesis unfolding b_def by arith
121         next
122           txt {* The second case is @{text "y = \<bar>f x\<bar> / \<parallel>x\<parallel>"} for some
123             @{text "x \<in> V"} with @{text "x \<noteq> 0"}. *}
124           assume "y \<noteq> 0"
125           with y obtain x where y_rep: "y = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
126               and x: "x \<in> V" and neq: "x \<noteq> 0"
127             by (auto simp add: B_def divide_inverse)
128           from x neq have gt: "0 < \<parallel>x\<parallel>" ..
130           txt {* The thesis follows by a short calculation using the
131             fact that @{text f} is bounded. *}
133           note y_rep
134           also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
135           proof (rule mult_right_mono)
136             from c x show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
137             from gt have "0 < inverse \<parallel>x\<parallel>"
138               by (rule positive_imp_inverse_positive)
139             then show "0 \<le> inverse \<parallel>x\<parallel>" by (rule order_less_imp_le)
140           qed
141           also have "\<dots> = c * (\<parallel>x\<parallel> * inverse \<parallel>x\<parallel>)"
142             by (rule Groups.mult_assoc)
143           also
144           from gt have "\<parallel>x\<parallel> \<noteq> 0" by simp
145           then have "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by simp
146           also have "c * 1 \<le> b" by (simp add: b_def le_maxI1)
147           finally show "y \<le> b" .
148         qed
149       qed
150       then show ?thesis ..
151     qed
152   qed
153   then show ?thesis unfolding fn_norm_def by (rule the_lubI_ex)
154 qed
156 lemma (in normed_vectorspace_with_fn_norm) fn_norm_ub [iff?]:
157   assumes "continuous V norm f"
158   assumes b: "b \<in> B V f"
159   shows "b \<le> \<parallel>f\<parallel>\<hyphen>V"
160 proof -
161   interpret continuous V norm f by fact
162   have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
163     using continuous V norm f by (rule fn_norm_works)
164   from this and b show ?thesis ..
165 qed
167 lemma (in normed_vectorspace_with_fn_norm) fn_norm_leastB:
168   assumes "continuous V norm f"
169   assumes b: "\<And>b. b \<in> B V f \<Longrightarrow> b \<le> y"
170   shows "\<parallel>f\<parallel>\<hyphen>V \<le> y"
171 proof -
172   interpret continuous V norm f by fact
173   have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
174     using continuous V norm f by (rule fn_norm_works)
175   from this and b show ?thesis ..
176 qed
178 text {* The norm of a continuous function is always @{text "\<ge> 0"}. *}
180 lemma (in normed_vectorspace_with_fn_norm) fn_norm_ge_zero [iff]:
181   assumes "continuous V norm f"
182   shows "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
183 proof -
184   interpret continuous V norm f by fact
185   txt {* The function norm is defined as the supremum of @{text B}.
186     So it is @{text "\<ge> 0"} if all elements in @{text B} are @{text "\<ge>
187     0"}, provided the supremum exists and @{text B} is not empty. *}
188   have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
189     using continuous V norm f by (rule fn_norm_works)
190   moreover have "0 \<in> B V f" ..
191   ultimately show ?thesis ..
192 qed
194 text {*
195   \medskip The fundamental property of function norms is:
196   \begin{center}
197   @{text "\<bar>f x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
198   \end{center}
199 *}
201 lemma (in normed_vectorspace_with_fn_norm) fn_norm_le_cong:
202   assumes "continuous V norm f" "linearform V f"
203   assumes x: "x \<in> V"
204   shows "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
205 proof -
206   interpret continuous V norm f by fact
207   interpret linearform V f by fact
208   show ?thesis
209   proof cases
210     assume "x = 0"
211     then have "\<bar>f x\<bar> = \<bar>f 0\<bar>" by simp
212     also have "f 0 = 0" by rule unfold_locales
213     also have "\<bar>\<dots>\<bar> = 0" by simp
214     also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
215       using continuous V norm f by (rule fn_norm_ge_zero)
216     from x have "0 \<le> norm x" ..
217     with a have "0 \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" by (simp add: zero_le_mult_iff)
218     finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" .
219   next
220     assume "x \<noteq> 0"
221     with x have neq: "\<parallel>x\<parallel> \<noteq> 0" by simp
222     then have "\<bar>f x\<bar> = (\<bar>f x\<bar> * inverse \<parallel>x\<parallel>) * \<parallel>x\<parallel>" by simp
223     also have "\<dots> \<le>  \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
224     proof (rule mult_right_mono)
225       from x show "0 \<le> \<parallel>x\<parallel>" ..
226       from x and neq have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<in> B V f"
227         by (auto simp add: B_def divide_inverse)
228       with continuous V norm f show "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>V"
229         by (rule fn_norm_ub)
230     qed
231     finally show ?thesis .
232   qed
233 qed
235 text {*
236   \medskip The function norm is the least positive real number for
237   which the following inequation holds:
238   \begin{center}
239     @{text "\<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
240   \end{center}
241 *}
243 lemma (in normed_vectorspace_with_fn_norm) fn_norm_least [intro?]:
244   assumes "continuous V norm f"
245   assumes ineq: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" and ge: "0 \<le> c"
246   shows "\<parallel>f\<parallel>\<hyphen>V \<le> c"
247 proof -
248   interpret continuous V norm f by fact
249   show ?thesis
250   proof (rule fn_norm_leastB [folded B_def fn_norm_def])
251     fix b assume b: "b \<in> B V f"
252     show "b \<le> c"
253     proof cases
254       assume "b = 0"
255       with ge show ?thesis by simp
256     next
257       assume "b \<noteq> 0"
258       with b obtain x where b_rep: "b = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
259         and x_neq: "x \<noteq> 0" and x: "x \<in> V"
260         by (auto simp add: B_def divide_inverse)
261       note b_rep
262       also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
263       proof (rule mult_right_mono)
264         have "0 < \<parallel>x\<parallel>" using x x_neq ..
265         then show "0 \<le> inverse \<parallel>x\<parallel>" by simp
266         from ineq and x show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
267       qed
268       also have "\<dots> = c"
269       proof -
270         from x_neq and x have "\<parallel>x\<parallel> \<noteq> 0" by simp
271         then show ?thesis by simp
272       qed
273       finally show ?thesis .
274     qed
275   qed (insert continuous V norm f, simp_all add: continuous_def)
276 qed
278 end