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theory Function_Norm(* Title: HOL/Hahn_Banach/Function_Norm.thy
Author: Gertrud Bauer, TU Munich
*)
header {* The norm of a function *}
theory Function_Norm
imports Normed_Space Function_Order
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 divide_inverse)
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 Groups.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 divide_inverse)
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 divide_inverse)
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