# Theory Hahn_Banach

theory Hahn_Banach
imports Hahn_Banach_Lemmas
(*  Title:      HOL/Hahn_Banach/Hahn_Banach.thy
Author:     Gertrud Bauer, TU Munich
*)

section ‹The Hahn-Banach Theorem›

theory Hahn_Banach
imports Hahn_Banach_Lemmas
begin

text ‹
We present the proof of two different versions of the Hahn-Banach Theorem,
closely following @{cite ‹\S36› "Heuser:1986"}.
›

subsection ‹The Hahn-Banach Theorem for vector spaces›

paragraph ‹Hahn-Banach Theorem.›
text ‹
Let ‹F› be a subspace of a real vector space ‹E›, let ‹p› be a semi-norm on
‹E›, and ‹f› be a linear form defined on ‹F› such that ‹f› is bounded by
‹p›, i.e. ‹∀x ∈ F. f x ≤ p x›. Then ‹f› can be extended to a linear form ‹h›
on ‹E› such that ‹h› is norm-preserving, i.e. ‹h› is also bounded by ‹p›.
›

paragraph ‹Proof Sketch.›
text ‹
▸ Define ‹M› as the set of norm-preserving extensions of ‹f› to subspaces of
‹E›. The linear forms in ‹M› are ordered by domain extension.

▸ We show that every non-empty chain in ‹M› has an upper bound in ‹M›.

▸ With Zorn's Lemma we conclude that there is a maximal function ‹g› in ‹M›.

▸ The domain ‹H› of ‹g› is the whole space ‹E›, as shown by classical

▪ Assuming ‹g› is not defined on whole ‹E›, it can still be extended in a
norm-preserving way to a super-space ‹H'› of ‹H›.

▪ Thus ‹g› can not be maximal. Contradiction!
›

theorem Hahn_Banach:
assumes E: "vectorspace E" and "subspace F E"
and "seminorm E p" and "linearform F f"
assumes fp: "∀x ∈ F. f x ≤ p x"
shows "∃h. linearform E h ∧ (∀x ∈ F. h x = f x) ∧ (∀x ∈ E. h x ≤ p x)"
― ‹Let ‹E› be a vector space, ‹F› a subspace of ‹E›, ‹p› a seminorm on ‹E›,›
― ‹and ‹f› a linear form on ‹F› such that ‹f› is bounded by ‹p›,›
― ‹then ‹f› can be extended to a linear form ‹h› on ‹E› in a norm-preserving way. ┈›
proof -
interpret vectorspace E by fact
interpret subspace F E by fact
interpret seminorm E p by fact
interpret linearform F f by fact
define M where "M = norm_pres_extensions E p F f"
then have M: "M = …" by (simp only:)
from E have F: "vectorspace F" ..
note FE = ‹F ⊴ E›
{
fix c assume cM: "c ∈ chains M" and ex: "∃x. x ∈ c"
have "⋃c ∈ M"
― ‹Show that every non-empty chain ‹c› of ‹M› has an upper bound in ‹M›:›
― ‹‹⋃c› is greater than any element of the chain ‹c›, so it suffices to show ‹⋃c ∈ M›.›
unfolding M_def
proof (rule norm_pres_extensionI)
let ?H = "domain (⋃c)"
let ?h = "funct (⋃c)"

have a: "graph ?H ?h = ⋃c"
proof (rule graph_domain_funct)
fix x y z assume "(x, y) ∈ ⋃c" and "(x, z) ∈ ⋃c"
with M_def cM show "z = y" by (rule sup_definite)
qed
moreover from M cM a have "linearform ?H ?h"
by (rule sup_lf)
moreover from a M cM ex FE E have "?H ⊴ E"
by (rule sup_subE)
moreover from a M cM ex FE have "F ⊴ ?H"
by (rule sup_supF)
moreover from a M cM ex have "graph F f ⊆ graph ?H ?h"
by (rule sup_ext)
moreover from a M cM have "∀x ∈ ?H. ?h x ≤ p x"
by (rule sup_norm_pres)
ultimately show "∃H h. ⋃c = graph H h
∧ linearform H h
∧ H ⊴ E
∧ F ⊴ H
∧ graph F f ⊆ graph H h
∧ (∀x ∈ H. h x ≤ p x)" by blast
qed
}
then have "∃g ∈ M. ∀x ∈ M. g ⊆ x ⟶ x = g"
― ‹With Zorn's Lemma we can conclude that there is a maximal element in ‹M›. ┈›
proof (rule Zorn's_Lemma)
― ‹We show that ‹M› is non-empty:›
show "graph F f ∈ M"
unfolding M_def
proof (rule norm_pres_extensionI2)
show "linearform F f" by fact
show "F ⊴ E" by fact
from F show "F ⊴ F" by (rule vectorspace.subspace_refl)
show "graph F f ⊆ graph F f" ..
show "∀x∈F. f x ≤ p x" by fact
qed
qed
then obtain g where gM: "g ∈ M" and gx: "∀x ∈ M. g ⊆ x ⟶ g = x"
by blast
from gM obtain H h where
g_rep: "g = graph H h"
and linearform: "linearform H h"
and HE: "H ⊴ E" and FH: "F ⊴ H"
and graphs: "graph F f ⊆ graph H h"
and hp: "∀x ∈ H. h x ≤ p x" unfolding M_def ..
― ‹‹g› is a norm-preserving extension of ‹f›, in other words:›
― ‹‹g› is the graph of some linear form ‹h› defined on a subspace ‹H› of ‹E›,›
― ‹and ‹h› is an extension of ‹f› that is again bounded by ‹p›. ┈›
from HE E have H: "vectorspace H"
by (rule subspace.vectorspace)

have HE_eq: "H = E"
― ‹We show that ‹h› is defined on whole ‹E› by classical contradiction. ┈›
proof (rule classical)
assume neq: "H ≠ E"
― ‹Assume ‹h› is not defined on whole ‹E›. Then show that ‹h› can be extended›
― ‹in a norm-preserving way to a function ‹h'› with the graph ‹g'›. ┈›
have "∃g' ∈ M. g ⊆ g' ∧ g ≠ g'"
proof -
from HE have "H ⊆ E" ..
with neq obtain x' where x'E: "x' ∈ E" and "x' ∉ H" by blast
obtain x': "x' ≠ 0"
proof
show "x' ≠ 0"
proof
assume "x' = 0"
with H have "x' ∈ H" by (simp only: vectorspace.zero)
with ‹x' ∉ H› show False by contradiction
qed
qed

define H' where "H' = H + lin x'"
― ‹Define ‹H'› as the direct sum of ‹H› and the linear closure of ‹x'›. ┈›
have HH': "H ⊴ H'"
proof (unfold H'_def)
from x'E have "vectorspace (lin x')" ..
with H show "H ⊴ H + lin x'" ..
qed

obtain xi where
xi: "∀y ∈ H. - p (y + x') - h y ≤ xi
∧ xi ≤ p (y + x') - h y"
― ‹Pick a real number ‹ξ› that fulfills certain inequality; this will›
― ‹be used to establish that ‹h'› is a norm-preserving extension of ‹h›.
\label{ex-xi-use}┈›
proof -
from H have "∃xi. ∀y ∈ H. - p (y + x') - h y ≤ xi
∧ xi ≤ p (y + x') - h y"
proof (rule ex_xi)
fix u v assume u: "u ∈ H" and v: "v ∈ H"
with HE have uE: "u ∈ E" and vE: "v ∈ E" by auto
from H u v linearform have "h v - h u = h (v - u)"
also from hp and H u v have "… ≤ p (v - u)"
by (simp only: vectorspace.diff_closed)
also from x'E uE vE have "v - u = x' + - x' + v + - u"
also from x'E uE vE have "… = v + x' + - (u + x')"
also from x'E uE vE have "… = (v + x') - (u + x')"
also from x'E uE vE E have "p … ≤ p (v + x') + p (u + x')"
finally have "h v - h u ≤ p (v + x') + p (u + x')" .
then show "- p (u + x') - h u ≤ p (v + x') - h v" by simp
qed
then show thesis by (blast intro: that)
qed

define h' where "h' x = (let (y, a) =
SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H in h y + a * xi)" for x
― ‹Define the extension ‹h'› of ‹h› to ‹H'› using ‹ξ›. ┈›

have "g ⊆ graph H' h' ∧ g ≠ graph H' h'"
― ‹‹h'› is an extension of ‹h› \dots ┈›
proof
show "g ⊆ graph H' h'"
proof -
have "graph H h ⊆ graph H' h'"
proof (rule graph_extI)
fix t assume t: "t ∈ H"
from E HE t have "(SOME (y, a). t = y + a ⋅ x' ∧ y ∈ H) = (t, 0)"
using ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› by (rule decomp_H'_H)
with h'_def show "h t = h' t" by (simp add: Let_def)
next
from HH' show "H ⊆ H'" ..
qed
with g_rep show ?thesis by (simp only:)
qed

show "g ≠ graph H' h'"
proof -
have "graph H h ≠ graph H' h'"
proof
assume eq: "graph H h = graph H' h'"
have "x' ∈ H'"
unfolding H'_def
proof
from H show "0 ∈ H" by (rule vectorspace.zero)
from x'E show "x' ∈ lin x'" by (rule x_lin_x)
from x'E show "x' = 0 + x'" by simp
qed
then have "(x', h' x') ∈ graph H' h'" ..
with eq have "(x', h' x') ∈ graph H h" by (simp only:)
then have "x' ∈ H" ..
with ‹x' ∉ H› show False by contradiction
qed
with g_rep show ?thesis by simp
qed
qed
moreover have "graph H' h' ∈ M"
― ‹and ‹h'› is norm-preserving. ┈›
proof (unfold M_def)
show "graph H' h' ∈ norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
show "linearform H' h'"
using h'_def H'_def HE linearform ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› E
by (rule h'_lf)
show "H' ⊴ E"
unfolding H'_def
proof
show "H ⊴ E" by fact
show "vectorspace E" by fact
from x'E show "lin x' ⊴ E" ..
qed
from H ‹F ⊴ H› HH' show FH': "F ⊴ H'"
by (rule vectorspace.subspace_trans)
show "graph F f ⊆ graph H' h'"
proof (rule graph_extI)
fix x assume x: "x ∈ F"
with graphs have "f x = h x" ..
also have "… = h x + 0 * xi" by simp
also have "… = (let (y, a) = (x, 0) in h y + a * xi)"
also have "(x, 0) =
(SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H)"
using E HE
proof (rule decomp_H'_H [symmetric])
from FH x show "x ∈ H" ..
from x' show "x' ≠ 0" .
show "x' ∉ H" by fact
show "x' ∈ E" by fact
qed
also have
"(let (y, a) = (SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H)
in h y + a * xi) = h' x" by (simp only: h'_def)
finally show "f x = h' x" .
next
from FH' show "F ⊆ H'" ..
qed
show "∀x ∈ H'. h' x ≤ p x"
using h'_def H'_def ‹x' ∉ H› ‹x' ∈ E› ‹x' ≠ 0› E HE
‹seminorm E p› linearform and hp xi
by (rule h'_norm_pres)
qed
qed
ultimately show ?thesis ..
qed
then have "¬ (∀x ∈ M. g ⊆ x ⟶ g = x)" by simp
― ‹So the graph ‹g› of ‹h› cannot be maximal. Contradiction! ┈›
with gx show "H = E" by contradiction
qed

from HE_eq and linearform have "linearform E h"
by (simp only:)
moreover have "∀x ∈ F. h x = f x"
proof
fix x assume "x ∈ F"
with graphs have "f x = h x" ..
then show "h x = f x" ..
qed
moreover from HE_eq and hp have "∀x ∈ E. h x ≤ p x"
by (simp only:)
ultimately show ?thesis by blast
qed

subsection ‹Alternative formulation›

text ‹
The following alternative formulation of the Hahn-Banach
Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear form ‹f›
and a seminorm ‹p› the following inequality are equivalent:\footnote{This
was shown in lemma @{thm [source] abs_ineq_iff} (see page
\pageref{abs-ineq-iff}).}
\begin{center}
\begin{tabular}{lll}
‹∀x ∈ H. ¦h x¦ ≤ p x› & and & ‹∀x ∈ H. h x ≤ p x› \\
\end{tabular}
\end{center}
›

theorem abs_Hahn_Banach:
assumes E: "vectorspace E" and FE: "subspace F E"
and lf: "linearform F f" and sn: "seminorm E p"
assumes fp: "∀x ∈ F. ¦f x¦ ≤ p x"
shows "∃g. linearform E g
∧ (∀x ∈ F. g x = f x)
∧ (∀x ∈ E. ¦g x¦ ≤ p x)"
proof -
interpret vectorspace E by fact
interpret subspace F E by fact
interpret linearform F f by fact
interpret seminorm E p by fact
have "∃g. linearform E g ∧ (∀x ∈ F. g x = f x) ∧ (∀x ∈ E. g x ≤ p x)"
using E FE sn lf
proof (rule Hahn_Banach)
show "∀x ∈ F. f x ≤ p x"
using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])
qed
then obtain g where lg: "linearform E g" and *: "∀x ∈ F. g x = f x"
and **: "∀x ∈ E. g x ≤ p x" by blast
have "∀x ∈ E. ¦g x¦ ≤ p x"
using _ E sn lg **
proof (rule abs_ineq_iff [THEN iffD2])
show "E ⊴ E" ..
qed
with lg * show ?thesis by blast
qed

subsection ‹The Hahn-Banach Theorem for normed spaces›

text ‹
Every continuous linear form ‹f› on a subspace ‹F› of a norm space ‹E›, can
be extended to a continuous linear form ‹g› on ‹E› such that ‹∥f∥ = ∥g∥›.
›

theorem norm_Hahn_Banach:
fixes V and norm ("∥_∥")
fixes B defines "⋀V f. B V f ≡ {0} ∪ {¦f x¦ / ∥x∥ | x. x ≠ 0 ∧ x ∈ V}"
fixes fn_norm ("∥_∥‐_" [0, 1000] 999)
defines "⋀V f. ∥f∥‐V ≡ ⨆(B V f)"
assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"
and linearform: "linearform F f" and "continuous F f norm"
shows "∃g. linearform E g
∧ continuous E g norm
∧ (∀x ∈ F. g x = f x)
∧ ∥g∥‐E = ∥f∥‐F"
proof -
interpret normed_vectorspace E norm by fact
interpret normed_vectorspace_with_fn_norm E norm B fn_norm
by (auto simp: B_def fn_norm_def) intro_locales
interpret subspace F E by fact
interpret linearform F f by fact
interpret continuous F f norm by fact
have E: "vectorspace E" by intro_locales
have F: "vectorspace F" by rule intro_locales
have F_norm: "normed_vectorspace F norm"
using FE E_norm by (rule subspace_normed_vs)
have ge_zero: "0 ≤ ∥f∥‐F"
by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm ‹continuous F f norm› , folded B_def fn_norm_def])
txt ‹We define a function ‹p› on ‹E› as follows:
‹p x = ∥f∥ ⋅ ∥x∥››
define p where "p x = ∥f∥‐F * ∥x∥" for x

txt ‹‹p› is a seminorm on ‹E›:›
have q: "seminorm E p"
proof
fix x y a assume x: "x ∈ E" and y: "y ∈ E"

txt ‹‹p› is positive definite:›
have "0 ≤ ∥f∥‐F" by (rule ge_zero)
moreover from x have "0 ≤ ∥x∥" ..
ultimately show "0 ≤ p x"

txt ‹‹p› is absolutely homogeneous:›

show "p (a ⋅ x) = ¦a¦ * p x"
proof -
have "p (a ⋅ x) = ∥f∥‐F * ∥a ⋅ x∥" by (simp only: p_def)
also from x have "∥a ⋅ x∥ = ¦a¦ * ∥x∥" by (rule abs_homogenous)
also have "∥f∥‐F * (¦a¦ * ∥x∥) = ¦a¦ * (∥f∥‐F * ∥x∥)" by simp
also have "… = ¦a¦ * p x" by (simp only: p_def)
finally show ?thesis .
qed

show "p (x + y) ≤ p x + p y"
proof -
have "p (x + y) = ∥f∥‐F * ∥x + y∥" by (simp only: p_def)
also have a: "0 ≤ ∥f∥‐F" by (rule ge_zero)
from x y have "∥x + y∥ ≤ ∥x∥ + ∥y∥" ..
with a have " ∥f∥‐F * ∥x + y∥ ≤ ∥f∥‐F * (∥x∥ + ∥y∥)"
also have "… = ∥f∥‐F * ∥x∥ + ∥f∥‐F * ∥y∥" by (simp only: distrib_left)
also have "… = p x + p y" by (simp only: p_def)
finally show ?thesis .
qed
qed

txt ‹‹f› is bounded by ‹p›.›

have "∀x ∈ F. ¦f x¦ ≤ p x"
proof
fix x assume "x ∈ F"
with ‹continuous F f norm› and linearform
show "¦f x¦ ≤ p x"
unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm, folded B_def fn_norm_def])
qed

txt ‹Using the fact that ‹p› is a seminorm and ‹f› is bounded by ‹p› we can
apply the Hahn-Banach Theorem for real vector spaces. So ‹f› can be
extended in a norm-preserving way to some function ‹g› on the whole vector
space ‹E›.›

with E FE linearform q obtain g where
linearformE: "linearform E g"
and a: "∀x ∈ F. g x = f x"
and b: "∀x ∈ E. ¦g x¦ ≤ p x"
by (rule abs_Hahn_Banach [elim_format]) iprover

txt ‹We furthermore have to show that ‹g› is also continuous:›

have g_cont: "continuous E g norm" using linearformE
proof
fix x assume "x ∈ E"
with b show "¦g x¦ ≤ ∥f∥‐F * ∥x∥"
by (simp only: p_def)
qed

txt ‹To complete the proof, we show that ‹∥g∥ = ∥f∥›.›

have "∥g∥‐E = ∥f∥‐F"
proof (rule order_antisym)
txt ‹
First we show ‹∥g∥ ≤ ∥f∥›. The function norm ‹∥g∥› is defined as the
smallest ‹c ∈ ℝ› such that
\begin{center}
\begin{tabular}{l}
‹∀x ∈ E. ¦g x¦ ≤ c ⋅ ∥x∥›
\end{tabular}
\end{center}
⇤ Furthermore holds
\begin{center}
\begin{tabular}{l}
‹∀x ∈ E. ¦g x¦ ≤ ∥f∥ ⋅ ∥x∥›
\end{tabular}
\end{center}
›

from g_cont _ ge_zero
show "∥g∥‐E ≤ ∥f∥‐F"
proof
fix x assume "x ∈ E"
with b show "¦g x¦ ≤ ∥f∥‐F * ∥x∥"
by (simp only: p_def)
qed

txt ‹The other direction is achieved by a similar argument.›

show "∥f∥‐F ≤ ∥g∥‐E"
proof (rule normed_vectorspace_with_fn_norm.fn_norm_least
[OF normed_vectorspace_with_fn_norm.intro,
OF F_norm, folded B_def fn_norm_def])
fix x assume x: "x ∈ F"
show "¦f x¦ ≤ ∥g∥‐E * ∥x∥"
proof -
from a x have "g x = f x" ..
then have "¦f x¦ = ¦g x¦" by (simp only:)
also from g_cont have "… ≤ ∥g∥‐E * ∥x∥"
proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])
from FE x show "x ∈ E" ..
qed
finally show ?thesis .
qed
next
show "0 ≤ ∥g∥‐E"
using g_cont by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
show "continuous F f norm" by fact
qed
qed
with linearformE a g_cont show ?thesis by blast
qed

end