Hahn-Banach theorems - Department of Mathematics, IIT Madras

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HAHN BANACH THEOREMS
S.H.KULKARNI
Abstract.
We give an overview of Hahn-Banach theorems.We
present the statements of these theorems alongwith some defini-
tions that are required to understand these statements and make
some comments about the relevance,applications etc.Proofs are
not given.
1.Introduction
The Hahn-Banach theorems are one of the three most important
and fundamental theorems in basic Functional Analysis,the other two
being the Uniform Boundedness Principle and the Closed Graph The-
orem.Usually Hahn-Banach theorems are taught before the other two
and most books also present Hahn-Banach theorems ahead of Uniform
Boundedness Principle or the Closed Graph Theorem.This may be
due to several reasons.The statements,proofs and applications of
Hahn-Banach theorems are relatively easier to understand.In par-
ticular,the hypotheses do not include completeness of the underlying
normed linear spaces and proofs do not involve the use of Baire Cat-
egory Theorem.In this article,
1
we give an overview of Hahn-Banach
theorems.We present the statements of these theorems alongwith some
definitions that are required to understand these statements and make
some comments about the relevance,applications etc.Detailed proofs
are not given as these can be found in any introductory textbook on
Functional Analysis for example [1,4,6].
There are two classes of theorems commonly known as Hahn-Banach
theorems,namely Hahn-Banach theorems in the extension form and
Hahn-Banach theorems in the separation form.All these theorems as-
sert the existence of a linear functional with certain properties.Why is
it important to know the existence of such functionals?In a large num-
ber of applications of practical importance,the objects of study can be
viewed as members of a vector space.A study of such objects involves
making various measurements/observations.These are functionals on
that vector space.As the names suggest,Hahn-Banach theorems in
the extension form assert that functionals defined on a subspace of a
vector space (frequently with some additional structure,usually with a
normor topology) and having some additional properties (like linearity,
1
An expanded version of a talk given at a Regional Workshop in Analysis,
Villupuram.
1
2 S.H.KULKARNI
continuity) can be extended to the whole space while retaining these
additional properties.This is useful in asserting the existence of certain
functionals and this in turn can be used in applications involving ap-
proximation of certain functions.These theorems and their proofs are
analytic in nature.On the other hand,Hahn-Banach theorems in the
separation form and also their proofs are geometric in nature.These
theorems deal with the following question:Given two disjoint convex
sets in a vector space,when is it possible to find a hyperplane such that
the two given convex sets lie on the opposite sides of that hyperplane?
This information is useful in answering several questions dealing with
optimization problems,also known as Mathematical Programming.
Hahn-Banach theorems in both these forms are mathematically equiv-
alent.This means that it is possible to prove a Hahn-Banach theorem
in the extension form first and then use it to prove a corresponding
theorem in the separation form or prove a Hahn-Banach theorem in
the separation form first and then use it to prove a corresponding the-
orem in the extension form.Preference among these two approaches
is a matter of personal taste.Each of these approaches can be found
in the text books,for example the first approach can be found in [1]
whereas the second can be found in [4].
2.Functionals and their extensions
Hahn-Banach theorems are essentially theorems about real vector
spaces.Basic theorems are first proved for real vector spaces.These
are then extended to the case of complex vector spaces by means of a
technical result.(See Lemma 7.1 of [4] and remarks preceding it.) In
this article,we shall confine ourselves to real vector spaces.Common
examples of real vector spaces are
R
n
,sequence spaces
l
p
,1 ≤ p ≤ ∞,
function spaces C([a,b]) with pointwise or coordinatewise operations as
the case may be.A functional on such a vector space is a real valued
function defined on it.Next we list several properties that such a
functional may or may not have.
Definition 2.1.
Let V be a real vector space.A functional on V is a
function
φ:V →R.Such a φ is said to be
(1)
additive if
φ(a +b) = φ(a) +φ(b)
for all a,b ∈ V,
(2)
homogeneous if
φ(αa) = αφ(a)
for all a ∈ V and
α ∈ R,
(3)
linear if it is additive and homogeneous,
(4)
subadditive if
φ(a +b) ≤ φ(a) +φ(b)
HAHN BANACH THEOREMS 3
for all a,b ∈ V,
(5)
positive homogeneous if
φ(αa) = αφ(a)
for all a ∈ V and
α ∈ R with α ≥ 0,
(6)
sublinear if it is subadditive and positive homogeneous,
(7)
convex if
φ((1 −t)a +tb) ≤ (1 −t)φ(a) +tφ(b)
for all a,b ∈ V and t ∈ [0,1].
Several examples of functionals having some of the above properties
and not having some of the other properties can be given.These are
easy to construct and can be found in most books on Functional Anal-
ysis,including [1,4,6].For example,the functional φ defined on R
2
by
φ(x
1
,x
2
):= |x
1
| +|x
2
|
for
(x
1
,x
2
) ∈ R
2
is sublinear but not linear.On the other hand,the
functional ψ defined on R
2
by
ψ(x
1
,x
2
):= 2x
1
+x
2
for
(x
1
,x
2
) ∈ R
2
is linear.It is obvious that a linear functional satisfies
all the above properties.It is a good exercise to construct examples of
functionals satisfying one of the above properties but are not linear.
We are now in a position to state the first of the Hahn Banach
Extension theorems.This is used in proving all the other extension
theorems.It says that a linear functional defined on a subspace of a
real vector space V and which is dominated by a sublinear functional
defined on V has a linear extension which is also dominated by the
same sublinear functional.
Theorem 2.2.
Let V be a real vector space and p be a sublinear func-
tional defined on V.Suppose W is a subspace of V and φ is a linear
functional defined on W such that φ(x) ≤ p(x) for all x ∈ W.Then
there exists a linear functional ψ defined on V such that ψ(x) = φ(x)
for all x ∈ W and
ψ(y) ≤ p(y) for all y ∈ V.
Usual proof of this theorem involves two steps.(See [6] for details.)
Of course,if W = V,there is nothing to prove.When W is a proper
subspace,some x
0
∈ V\W is chosen and φ is extended to the lin-
ear span W
0
of W ∪ {x
0
} in such a way that the extension ψ remains
dominated by p in W
0
.This is the first step.The crucial idea in this
step is to choose the value of ψ(x
0
) in an appropriate manner.The
second step makes use of Zorn’s Lemma to construct a maximal sub-
space containing W to which φ can be extended satisfying the required
condition.Finally it is shown using the first step that this maximal
subspace must coincide with V.
4 S.H.KULKARNI
Though the above formulation of the Hahn Banach Extension the-
orem is most basic,the most popular one is stated in the context of
normed linear spaces.In order to understand this version,we need to
review a few basic definitions.
Definition 2.3.
Let X be a real vector space.A norm on X is a
function
￿.￿:X →R satisfying:
(1)
￿x￿ ≥ 0 for all x ∈ X and ￿x￿ = 0 if and only if
x = 0.
(2)
￿αx￿ = |α|￿x￿ for all x ∈ X and
α ∈ R.
(3)
￿x +y￿ ≤ ￿x￿ +￿y￿ for all x,y ∈ X.
A real normed linear space is a pair (X,￿.￿) where X is a real vector
space and
￿.￿ is a norm on X.
Examples of real normed linear spaces include
R
n
(with several norms),
sequence spaces l
p
,1 ≤ p ≤ ∞,function spaces
L
p
,1 ≤ p ≤ ∞,
C(X),
C([a,b]),C
1
([a,b]).
Among the various norms on R
n
,the following are more frequently
used.For
x = (x
1
,...,x
n
) ∈ R
n
,
(1)
￿x￿
1
:=
n
X
j=1
|x
j
|,
(2)
￿x￿
2
:= (
n
X
j=1
|x
j
|
2
)
1/2
,
(3)
￿x￿

:= max
j=1...n
|x
j
|.
Definition 2.4.
Let (X,￿.￿) be a normed linear space.A linear func-
tional φ on X is said to be bounded if
sup{|φ(x)|:x ∈ X,￿x￿ ≤ 1}
is finite.When this is the case,the above quantity is called the norm
of
φ and denoted by ￿φ￿.
It is easy to show (and is also proved in most textbooks) that a
linear functional φ on X is bounded if and only if it is continuous at
each x ∈ X if and only if it is uniformly continuous on X.Also note
that if φ is a bounded linear functional on X,then
|φ(x)| ≤ ￿φ￿￿x￿ for all x ∈ X.
See the references at the end,for several examples of bounded as well
as unbounded linear functionals.The popular version of the Hahn
Banach Extension theorem mentioned above says that every bounded
HAHN BANACH THEOREMS 5
linear functional defined on a subspace of a normed linear space has a
norm preserving extension to the whole space.
Theorem 2.5.
Let (X,￿.￿) be a normed linear space,Y a subspace of
X and φ a bounded linear functional defined on Y.Then there exists
a bounded linear functional ψ defined on X such that
φ(y) = ψ(y) for all y ∈ Y and ￿ψ￿ = ￿φ￿.
This is proved by observing that the functional p defined on X by
p(x):= ￿φ￿￿x￿,x ∈ X,is a sublinear functional that dominates φ
on Y and using Theorem 2.2.
This theoremis frequently used to deal with questions about approx-
imations.Typically we want to know whether an element a ∈ X can be
approximated by elements in a subspace Y,that is whether a belongs
to the closure of Y.For example one may want to know whether a con-
tinuous function f defined on the interval [0,1] can be approximated
by polynomials.This amounts to taking X = C([0,1]) and Y to be the
subspace of all polynomials.The following well known corollary of the
Hahn Banach Extension theorem helps in answering this question.
Corollary 2.6.
Let (X,￿.￿) be a normed linear space,Y be a subspace
of X and a ∈ X.Then a belongs to the closure of Y if and only if there
exists no bounded linear functional φ on X such that φ(y) = 0 for all
y ∈ Y and φ(a) ￿= 0.Equivalently,a does not belong to the closure of
Y if and only if there exists a bounded linear functional φ on X such
that φ(y) = 0 for all y ∈ Y and φ(a) ￿= 0.
Given X,Y and a how does one decide whether such a bounded
linear functional exists or not?In general,this is difficult to answer.
In order to answer this,we need to have the knowledge of all bounded
linear functionals on X.This is a theme of another class of theorems in
Functional Analysis known as Riesz Representation Theorems.These
theorems give complete description of bounded linear functionals on
various concrete normed linear spaces.See the references at the end
for more information on these theorems.An interesting application of
this idea to the Poisson Integral Formula can be found in [5].
3.Convex sets and their separation
In this section we discuss Hahn-Banach theorems in the separation
form.As mentioned in the introduction,these theorems deal with the
question of existence of a hyperplane that separates two given disjoint
convex subsets.We first need to review a few definitions and notations.
Let V be a real vector space,A,B be nonempty subsets of V and
α,a real number.Then the symbols A+B and αA have their natural
meaning as follows:
A+B:= {a +b:a ∈ A,b ∈ B}
6 S.H.KULKARNI
αA:= {αa:a ∈ A}.
Note that with these notations,
A+B = B +A
but A+A may not be the same as
2A.A special case of the above that
occurs very frequently is when B is a singleton,say {x}.Thus A+x
or x +A is the set
{x +a:a ∈ A}.
This is called the translate of
A by x for obvious reasons.A subset C
of V is said to be convex if for each t with 0 ≤ t ≤ 1,we have
(1 −t)C +tC ⊆ C.
In other words,for each
x,y ∈ C and 0 ≤ t ≤ 1,
(1 −t)x +ty ∈ C.
This is usually expressed by saying that if C contains two points,then it
should also contain the line segment joining those two points.We come
across several examples of convex sets.For example,every subspace is
a convex set.A translate of a subspace by a nonzero element is not
a subspace,but it is a convex set.Convex sets and functionals are
closely related concepts in the sense that we can associate convex sets
with functionals and vice versa.For example,if φ is a convex functional
(See Definition 2.1) and α is a real number,then the set
{x ∈ V:φ(x) ≤ α}
is a convex set.If φ is a linear functional,then each of the following is
a convex set.
(1)
{x ∈ V:φ(x) ≤ α},
(2)
{x ∈ V:φ(x) ≥ α},
(3)
{x ∈ V:φ(x) = α}.
Next recall that a subspace is called proper if it is different from the
whole space V.A hyperspace is a maximal proper subspace,that is a
proper subspace not properly contained in any other proper subspace.
If V is of finite dimension,say n,then every subspace of dimension
n −1 is a hyperspace.Thus in R
2
,every straight line passing through
the origin is a hyperspace.Similarly,hyperspaces in R
3
are the planes
passing through the origin.It is well known and is also easy to prove
that if φ is a nonzero linear functional on V,then the null space of φ
defined by
N(φ):= {x ∈ V:φ(x) = 0}
is a hyperspace.Also every hyperspace is a null space of some nonzero
linear functional.Next,a hyperplane is a translate of a hyperspace.
HAHN BANACH THEOREMS 7
Thus hyperplanes in
R
3
are the planes.(This may very well be the
reason for the name “hyperplane”.) Further,if H is a hyperplane,
then there exists a nonzero linear functional φ and a vector a ∈ V such
that
H = a +N(φ).
Now suppose φ(a) = α.Let h ∈ H.Then h = a+y for some y ∈ N(φ).
Hence
φ(h) = φ(a) +φ(y) = α.
On the other hand,suppose x ∈ V is such that φ(x) = α.Let y = x−a.
Then φ(y) = 0,that is y ∈ N(φ) and
x = a +y ∈ a +N(φ) = H.
Thus we have proved that with every hyperplane H,we can associate
a pair (φ,α),where φ is a nonzero linear functional and α is a real
number such that
H = {x ∈ V:φ(x) = α}.
For example,consider a plane given by the equation:
2x
1
−x
2
+7x
3
−5 = 0
in R
3
.Note that this is a hyperplane and the associated pair is (φ,5)
where the functional φ is given by
φ(x):= 2x
1
−x
2
+7x
3
,x = (x
1
,x
2
,x
3
) ∈ R
3
.
Every such hyperplane H divides the whole space V into two convex
sets:
H
l
= {x ∈ V:φ(x) ≤ α}
and
H
r
= {x ∈ V:φ(x) ≥ α}
known as halfspaces.We say that two nonempty subsets
A and B of
V are separated by H if A and B lie on different sides of H,that is A
and B are contained in different half spaces formed by H.
We are now ready to present the basic separation theorem.
Theorem 3.1.
Let A and B be nonempty disjoint convex subsets of a
real vector space V.Then A and B can be separated by a hyperplane,
that is,there exists a nonzero linear functional φ and a real number α
such that φ(x) ≤ α ≤ φ(y) for all x ∈ A and y ∈ B.
A proof of this can be found in [1].Also note that we can not think
of separating nonconvex sets by hyperplanes in this way.For example,
suppose A is a circle in R
2
with the centre at the origin and radius 1
and B is the singleton set consisting of the origin.Then no straight
line can separate A and B.Recall that straight lines are hyperplanes
in R
2
.
Again as in the case of extension theorems,though the above the-
orem is very basic,the more popular ones are stated (and proved) in
8 S.H.KULKARNI
the context of a normed linear space.(See Definition 2.3).Here we
are looking for separation of convex sets not by just arbitrary hyper-
planes but by closed hyperplanes.This means that the associated linear
functionals must be continuous.(See Definition 2.4 and the remarks
following it.) The following is usually known as the Hahn Banach Sep-
aration Theorem in normed linear spaces.Its proof can be found in
[1,4,6].
Theorem 3.2.
Let A and B be nonempty disjoint convex subsets of a
real normed linear space X.Suppose A has an interior point.Then A
and B can be separated by a closed hyperplane,that is,there exists a
nonzero continuous linear functional φ and a real number α such that
φ(x) < α ≤ φ(y) for all x ∈ A and y ∈ B.
The following Corollary of this theorem is also very popular and is
used in several applications.
Corollary 3.3.
Let A and B be nonempty disjoint convex subsets of
a real normed linear space X.Suppose A is compact and B is closed.
Then A and B can be strictly separated by a closed hyperplane,that is,
there exists a nonzero continuous linear functional φ and real numbers
α and β such that φ(x) < α < β < φ(y) for all x ∈ A and y ∈ B.
As mentioned in the introduction,these separation theorems have
applications to the problems of optimization,in particular to a class of
problems known as convex programming problems.We only mention
a very interesting and famous result known as Farkas’ Lemma which
is a direct consequence of the separation theorem.
Theorem 3.4.
Let A be a matrix of order m×n with real entries and
b ∈ R
m
,regarded as a column matrix of order m×1.Then exactly one
of the following alternatives hold.
(i) The system of equations
Ax = b has a nonnegative solution x ∈ R
n
.
(Here nonnegative means x
j
≥ 0 for each j = 1,...,n.)
(ii) The system of inequalities
y
T
A ≥ 0 and y
T
b < 0 has a solution
y ∈ R
m
.
It is easy to see that both the alternatives can not hold simultane-
ously,because if they do,then there exist x as in (i) and y as in (ii).
Then
0 ≤ (y
T
A)x = y
T
(Ax) = y
T
b < 0,
a contradiction.To prove that one of the alternatives must hold,the
convex set
C:= {Ax:x ∈ R
n
,x ≥ 0} ⊆ R
m
is considered.If
b ∈ C,the alternative (i) holds.Otherwise,C and {b}
are nonempty disjoint convex sets in R
m
and hence can be separated
by a hyperplane H.A proof consists of noting that with every such
HAHN BANACH THEOREMS 9
hyperplane H in R
m
,we can associate a vector y ∈ R
m
and a real
number α such that
H = {u ∈ R
m
:y
T
u = α}
and then showing that in this particular case α can be taken to be 0.
Details of this proof as well as the application of Farkas’ Lemma to the
duality theory of Linear Programming can be found in [2].A highly
refreshing and readable account of Farkas’ Lemma and its connection
with several problems in Mathematics and Economics is given in [3].
4.Limitations
After having said so many things about the importance,relevance
etc.of Hahn Banach theorems,we finally also point out some limita-
tions of these theorems and also of methods based on these theorems.
In fact,these limitations are common to many theorems and methods
of Functional Analysis.Statements of all these Hahn Banach theorems
are existence statements,that is,these statements assert the existence
of some linear functionals.However all the proofs are nonconstructive
in nature.This means that these proofs give no clue about how to find
the linear functional whose existence is asserted by a theorem,even
when the vector space under consideration is finite dimensional.(The
situation is very similar to various proofs of the Fundamental Theo-
rem of Algebra,none of which say anything about how to find a root
of a given polynomial,though the Theorem asserts that every such
polynomial must have a root.) For example,suppose we are given two
finite sets in R
10
.One way of giving these sets would be to give two
matrices each having ten rows.Let A and B denote the convex hulls
of these two sets.(Recall that a convex hull of a set is the smallest
convex set containing the given set.) Suppose we consider the follow-
ing problem:To determine whether A and B are disjoint and in case
these are disjoint,to find a hyperplane in R
10
separating A and B.It
may appear at first that the Hahn Banach Separation Theorem may
be useful to tackle this problem.It is useful in the sense that if A and
B are disjoint,the Theorem says that there exists a hyperplane in R
10
separating A and B.However it says nothing about how to find such
a hyperplane.Completely different methods have to be used to tackle
this problem.(See [2].)
References
[1]
B.Bollob´as,Linear analysis,Cambridge Univ.Press,Cambridge,1990.
MR1087297 (92a:46001)
[2]
J.Franklin,Methods of mathematical economics,Springer,New York,1980.
MR0602694 (82e:90002)
[3]
J.Franklin,Mathematical methods of economics,Amer.Math.Monthly 90
(1983),no.4,229–244.MR0700264 (84e:90001)
10 S.H.KULKARNI
[4]
B.V.Limaye,Functional analysis,Second edition,New Age,New Delhi,1996.
MR1427262 (97k:46001)
[5]
W.Rudin,Real and complex analysis,Third edition,McGraw-Hill,New York,
1987.MR0924157 (88k:00002)
[6]
A.E.Taylor and D.C.Lay,Introduction to functional analysis,Second edition,
John Wiley & Sons,New York,1980.MR0564653 (81b:46001)
Department of Mathematics,Indian Institute of Technology - Madras,
Chennai 600036
E-mail address:shk@iitm.ac.in