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theory FunctionNorm(* Title: HOL/Real/HahnBanach/FunctionNorm.thy Author: Gertrud Bauer, TU Munich *) header {* The norm of a function *} theory FunctionNorm imports NormedSpace FunctionOrder begin subsection {* Continuous linear forms*} text {* A linear form @{text f} on a normed vector space @{text "(V, \<parallel>·\<parallel>)"} is \emph{continuous}, iff it is bounded, i.e. \begin{center} @{text "∃c ∈ R. ∀x ∈ V. ¦f x¦ ≤ c · \<parallel>x\<parallel>"} \end{center} In our application no other functions than linear forms are considered, so we can define continuous linear forms as bounded linear forms: *} locale continuous = var_V + norm_syntax + linearform + assumes bounded: "∃c. ∀x ∈ V. ¦f x¦ ≤ c * \<parallel>x\<parallel>" declare continuous.intro [intro?] continuous_axioms.intro [intro?] lemma continuousI [intro]: fixes norm :: "_ => real" ("\<parallel>_\<parallel>") assumes "linearform V f" assumes r: "!!x. x ∈ V ==> ¦f x¦ ≤ c * \<parallel>x\<parallel>" shows "continuous V norm f" proof show "linearform V f" by fact from r have "∃c. ∀x∈V. ¦f x¦ ≤ c * \<parallel>x\<parallel>" by blast then show "continuous_axioms V norm f" .. qed subsection {* The norm of a linear form *} text {* The least real number @{text c} for which holds \begin{center} @{text "∀x ∈ V. ¦f x¦ ≤ c · \<parallel>x\<parallel>"} \end{center} is called the \emph{norm} of @{text f}. For non-trivial vector spaces @{text "V ≠ {0}"} the norm can be defined as \begin{center} @{text "\<parallel>f\<parallel> = \<sup>x ≠ 0. ¦f x¦ / \<parallel>x\<parallel>"} \end{center} For the case @{text "V = {0}"} the supremum would be taken from an empty set. Since @{text \<real>} is unbounded, there would be no supremum. To avoid this situation it must be guaranteed that there is an element in this set. This element must be @{text "{} ≥ 0"} so that @{text fn_norm} has the norm properties. Furthermore it does not have to change the norm in all other cases, so it must be @{text 0}, as all other elements are @{text "{} ≥ 0"}. Thus we define the set @{text B} where the supremum is taken from as follows: \begin{center} @{text "{0} ∪ {¦f x¦ / \<parallel>x\<parallel>. x ≠ 0 ∧ x ∈ F}"} \end{center} @{text fn_norm} is equal to the supremum of @{text B}, if the supremum exists (otherwise it is undefined). *} locale fn_norm = norm_syntax + fixes B defines "B V f ≡ {0} ∪ {¦f x¦ / \<parallel>x\<parallel> | x. x ≠ 0 ∧ x ∈ V}" fixes fn_norm ("\<parallel>_\<parallel>_" [0, 1000] 999) defines "\<parallel>f\<parallel>V ≡ \<Squnion>(B V f)" locale normed_vectorspace_with_fn_norm = normed_vectorspace + fn_norm lemma (in fn_norm) B_not_empty [intro]: "0 ∈ B V f" by (simp add: B_def) text {* The following lemma states that every continuous linear form on a normed space @{text "(V, \<parallel>·\<parallel>)"} has a function norm. *} lemma (in normed_vectorspace_with_fn_norm) fn_norm_works: assumes "continuous V norm f" shows "lub (B V f) (\<parallel>f\<parallel>V)" proof - interpret continuous V norm f by fact txt {* The existence of the supremum is shown using the completeness of the reals. Completeness means, that every non-empty bounded set of reals has a supremum. *} have "∃a. lub (B V f) a" proof (rule real_complete) txt {* First we have to show that @{text B} is non-empty: *} have "0 ∈ B V f" .. then show "∃x. x ∈ B V f" .. txt {* Then we have to show that @{text B} is bounded: *} show "∃c. ∀y ∈ B V f. y ≤ c" proof - txt {* We know that @{text f} is bounded by some value @{text c}. *} from bounded obtain c where c: "∀x ∈ V. ¦f x¦ ≤ c * \<parallel>x\<parallel>" .. txt {* To prove the thesis, we have to show that there is some @{text b}, such that @{text "y ≤ b"} for all @{text "y ∈ B"}. Due to the definition of @{text B} there are two cases. *} def b ≡ "max c 0" have "∀y ∈ B V f. y ≤ b" proof fix y assume y: "y ∈ B V f" show "y ≤ b" proof cases assume "y = 0" then show ?thesis unfolding b_def by arith next txt {* The second case is @{text "y = ¦f x¦ / \<parallel>x\<parallel>"} for some @{text "x ∈ V"} with @{text "x ≠ 0"}. *} assume "y ≠ 0" with y obtain x where y_rep: "y = ¦f x¦ * inverse \<parallel>x\<parallel>" and x: "x ∈ V" and neq: "x ≠ 0" by (auto simp add: B_def real_divide_def) from x neq have gt: "0 < \<parallel>x\<parallel>" .. txt {* The thesis follows by a short calculation using the fact that @{text f} is bounded. *} note y_rep also have "¦f x¦ * inverse \<parallel>x\<parallel> ≤ (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>" proof (rule mult_right_mono) from c x show "¦f x¦ ≤ c * \<parallel>x\<parallel>" .. from gt have "0 < inverse \<parallel>x\<parallel>" by (rule positive_imp_inverse_positive) then show "0 ≤ inverse \<parallel>x\<parallel>" by (rule order_less_imp_le) qed also have "… = c * (\<parallel>x\<parallel> * inverse \<parallel>x\<parallel>)" by (rule real_mult_assoc) also from gt have "\<parallel>x\<parallel> ≠ 0" by simp then have "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by simp also have "c * 1 ≤ b" by (simp add: b_def le_maxI1) finally show "y ≤ b" . qed qed then show ?thesis .. qed qed then show ?thesis unfolding fn_norm_def by (rule the_lubI_ex) qed lemma (in normed_vectorspace_with_fn_norm) fn_norm_ub [iff?]: assumes "continuous V norm f" assumes b: "b ∈ B V f" shows "b ≤ \<parallel>f\<parallel>V" proof - interpret continuous V norm f by fact have "lub (B V f) (\<parallel>f\<parallel>V)" using `continuous V norm f` by (rule fn_norm_works) from this and b show ?thesis .. qed lemma (in normed_vectorspace_with_fn_norm) fn_norm_leastB: assumes "continuous V norm f" assumes b: "!!b. b ∈ B V f ==> b ≤ y" shows "\<parallel>f\<parallel>V ≤ y" proof - interpret continuous V norm f by fact have "lub (B V f) (\<parallel>f\<parallel>V)" using `continuous V norm f` by (rule fn_norm_works) from this and b show ?thesis .. qed text {* The norm of a continuous function is always @{text "≥ 0"}. *} lemma (in normed_vectorspace_with_fn_norm) fn_norm_ge_zero [iff]: assumes "continuous V norm f" shows "0 ≤ \<parallel>f\<parallel>V" proof - interpret continuous V norm f by fact txt {* The function norm is defined as the supremum of @{text B}. So it is @{text "≥ 0"} if all elements in @{text B} are @{text "≥ 0"}, provided the supremum exists and @{text B} is not empty. *} have "lub (B V f) (\<parallel>f\<parallel>V)" using `continuous V norm f` by (rule fn_norm_works) moreover have "0 ∈ B V f" .. ultimately show ?thesis .. qed text {* \medskip The fundamental property of function norms is: \begin{center} @{text "¦f x¦ ≤ \<parallel>f\<parallel> · \<parallel>x\<parallel>"} \end{center} *} lemma (in normed_vectorspace_with_fn_norm) fn_norm_le_cong: assumes "continuous V norm f" "linearform V f" assumes x: "x ∈ V" shows "¦f x¦ ≤ \<parallel>f\<parallel>V * \<parallel>x\<parallel>" proof - interpret continuous V norm f by fact interpret linearform V f by fact show ?thesis proof cases assume "x = 0" then have "¦f x¦ = ¦f 0¦" by simp also have "f 0 = 0" by rule unfold_locales also have "¦…¦ = 0" by simp also have a: "0 ≤ \<parallel>f\<parallel>V" using `continuous V norm f` by (rule fn_norm_ge_zero) from x have "0 ≤ norm x" .. with a have "0 ≤ \<parallel>f\<parallel>V * \<parallel>x\<parallel>" by (simp add: zero_le_mult_iff) finally show "¦f x¦ ≤ \<parallel>f\<parallel>V * \<parallel>x\<parallel>" . next assume "x ≠ 0" with x have neq: "\<parallel>x\<parallel> ≠ 0" by simp then have "¦f x¦ = (¦f x¦ * inverse \<parallel>x\<parallel>) * \<parallel>x\<parallel>" by simp also have "… ≤ \<parallel>f\<parallel>V * \<parallel>x\<parallel>" proof (rule mult_right_mono) from x show "0 ≤ \<parallel>x\<parallel>" .. from x and neq have "¦f x¦ * inverse \<parallel>x\<parallel> ∈ B V f" by (auto simp add: B_def real_divide_def) with `continuous V norm f` show "¦f x¦ * inverse \<parallel>x\<parallel> ≤ \<parallel>f\<parallel>V" by (rule fn_norm_ub) qed finally show ?thesis . qed qed text {* \medskip The function norm is the least positive real number for which the following inequation holds: \begin{center} @{text "¦f x¦ ≤ c · \<parallel>x\<parallel>"} \end{center} *} lemma (in normed_vectorspace_with_fn_norm) fn_norm_least [intro?]: assumes "continuous V norm f" assumes ineq: "∀x ∈ V. ¦f x¦ ≤ c * \<parallel>x\<parallel>" and ge: "0 ≤ c" shows "\<parallel>f\<parallel>V ≤ c" proof - interpret continuous V norm f by fact show ?thesis proof (rule fn_norm_leastB [folded B_def fn_norm_def]) fix b assume b: "b ∈ B V f" show "b ≤ c" proof cases assume "b = 0" with ge show ?thesis by simp next assume "b ≠ 0" with b obtain x where b_rep: "b = ¦f x¦ * inverse \<parallel>x\<parallel>" and x_neq: "x ≠ 0" and x: "x ∈ V" by (auto simp add: B_def real_divide_def) note b_rep also have "¦f x¦ * inverse \<parallel>x\<parallel> ≤ (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>" proof (rule mult_right_mono) have "0 < \<parallel>x\<parallel>" using x x_neq .. then show "0 ≤ inverse \<parallel>x\<parallel>" by simp from ineq and x show "¦f x¦ ≤ c * \<parallel>x\<parallel>" .. qed also have "… = c" proof - from x_neq and x have "\<parallel>x\<parallel> ≠ 0" by simp then show ?thesis by simp qed finally show ?thesis . qed qed (insert `continuous V norm f`, simp_all add: continuous_def) qed end