The Hahn–Banach theorem is a central tool in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
It allows the extension of
bounded linear functionals defined on a subspace of some
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
to the whole space, and it also shows that there are "enough"
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear functionals defined on every
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
to make the study of the
dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the
hyperplane separation theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least on ...
, and has numerous uses in
convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...
.
History
The theorem is named for the mathematicians
Hans Hahn and
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...
, who proved it independently in the late
1920s
File:1920s decade montage.png, From left, clockwise: Third Tipperary Brigade Flying Column No. 2 under Seán Hogan during the Irish War of Independence; Prohibition agents destroying barrels of alcohol in accordance to the 18th amendment, whic ...
.
The special case of the theorem for the space
and
C( , b where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals. In effect, they needed to solve the following problem:
:() Given a collection
\left(f_i\right)_ of bounded linear functionals on a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
X and a collection of scalars
\left(c_i\right)_, determine if there is an
x \in X such that
f_i(x) = c_i for all
i \in I.
If
X happens to be a
reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
then to solve the vector problem, it suffices to solve the following dual problem:
:(The functional problem) Given a collection
\left(x_i\right)_ of vectors in a normed space
X and a collection of scalars
\left(c_i\right)_, determine if there is a bounded linear functional
f on
X such that
f\left(x_i\right) = c_i for all
i \in I.
Riesz went on to define
L^p( space">, 1 space (
1 < p < \infty) in
1910 and the
\ell^p spaces in
1913
Events January
* January 5 – First Balkan War: Battle of Lemnos (1913), Battle of Lemnos – Greek admiral Pavlos Kountouriotis forces the Turkish fleet to retreat to its base within the Dardanelles, from which it will not ven ...
. While investigating these spaces he proved a special case of the Hahn–Banach theorem. Helly also proved a special case of the Hahn–Banach theorem in 1912. In 1910, Riesz solved the functional problem for some specific spaces and in 1912, Helly solved it for a more general class of spaces. It wasn't until
1932
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* January 9 – Sakuradamon Incident (1932), Sakuradamon Incident: Korean nationalist Lee Bong-chang fails in his effort ...
that Banach, in one of the first important applications of the Hahn–Banach theorem, solved the general functional problem. The following theorem states the general functional problem and characterizes its solution.
The Hahn–Banach theorem can be deduced from the above theorem. If
X is
reflexive then this theorem solves the vector problem.
Hahn–Banach theorem
A real-valued function
f : M \to \R defined on a subset
M of
X is said to be a function
p : X \to \R if
f(m) \leq p(m) for every
m \in M.
Hence the reason why the following version of the Hahn-Banach theorem is called .
The theorem remains true if the requirements on
p are relaxed to require only that
p be a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
:
p(t x + (1 - t) y) \leq t p(x) + (1 - t) p(y) \qquad \text 0 < t < 1 \text x, y \in X.
A function
p : X \to \R is convex and satisfies
p(0) \leq 0 if and only if
p(a x + b y) \leq a p(x) + b p(y) for all vectors
x, y \in X and all non-negative real
a, b \geq 0 such that
a + b \leq 1.
Every
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
is a convex function. On the other hand, if
p : X \to \R is convex, the function
p_0\leq p defined by
p_0(x):=\inf_ \frac is sublinear, and satisfies
F\le p_0 for all linear functional
F\le p. So the extension of the Hahn-Banach theorem to convex functionals is easy, but has not a much larger content than the classical one stated for sublinear functionals.
If
F : X \to \R is linear then
F \leq p if and only if
-p(-x) \leq F(x) \leq p(x) \quad \text x \in X,
which is the (equivalent) conclusion that some authors write instead of
F \leq p.
It follows that if
p : X \to \R is also , meaning that
p(-x) = p(x) holds for all
x \in X, then
F \leq p if and only
, F, \leq p.
Every
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
and both are symmetric
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
sublinear functions. A sublinear function is a seminorm if and only if it is a
balanced function
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
. On a real vector space (although not on a complex vector space), a sublinear function is a seminorm if and only if it is symmetric. The
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
\R \to \R on
X := \R is an example of a sublinear function that is not a seminorm.
For complex or real vector spaces
The dominated extension theorem for real linear functionals implies the following alternative statement of the Hahn–Banach theorem that can be applied to linear functionals on real or complex vector spaces.
The theorem remains true if the requirements on
p are relaxed to require only that for all
x, y \in X and all scalars
a and
b satisfying
, a, + , b, \leq 1,
p(a x + b y) \leq , a, p(x) + , b, p(y).
This condition holds if and only if
p is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
and
balanced function
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \le ...
satisfying
p(0) \leq 0, or equivalently, if and only if it is convex, satisfies
p(0) \leq 0, and
p(u x) \leq p(x) for all
x \in X and all
unit length
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
scalars
u.
A complex-valued functional
F is said to be if
, F(x), \leq p(x) for all
x in the domain of
F.
With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly:
:Hahn–Banach dominated extension theorem: If
p : X \to \R is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
defined on a real or complex vector space
X, then every
dominated linear functional defined on a vector subspace of
X has a dominated linear extension to all of
X. In the case where
X is a real vector space and
p : X \to \R is merely a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
or
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
, this conclusion will remain true if both instances of "
dominated" (meaning
, F, \leq p) are weakened to instead mean "
dominated " (meaning
F \leq p).
Proof
The following observations allow the
Hahn–Banach theorem for real vector spaces to be applied to (complex-valued) linear functionals on complex vector spaces.
Every linear functional
F : X \to \Complex on a complex vector space is
completely determined by its
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
\; \operatorname F : X \to \R \; through the formula
[If z = a + i b \in \Complex has real part \operatorname z = a then - \operatorname (i z) = b, which proves that z = \operatorname z - i \operatorname (i z). Substituting F(x) in for z and using i F(x) = F(i x) gives F(x) = \operatorname F(x) - i \operatorname F(i x). \blacksquare]
F(x) \;=\; \operatorname F(x) - i \operatorname F(i x) \qquad \text x \in X
and moreover, if
\, \cdot\, is a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
on
X then their
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
s are equal:
\, F\, = \, \operatorname F\, .
In particular, a linear functional on
X extends another one defined on
M \subseteq X if and only if their real parts are equal on
M (in other words, a linear functional
F extends
f if and only if
\operatorname F extends
\operatorname f).
The real part of a linear functional on
X is always a real-linear functional (meaning that it is linear when
X is considered as a real vector space) and if
R : X \to \R is a real-linear functional on a complex vector space then
x \mapsto R(x) - i R(i x) defines the unique linear functional on
X whose real part is
R.
If
F is a linear functional on a (complex or real) vector space
X and if
p : X \to \R is a seminorm then
[Let F be any ]homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
scalar-valued map on X (such as a linear functional) and let p : X \to \R be any map that satisfies p(u x) = p(x) for all x and unit length
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
scalars u (such as a seminorm). If , F, \leq p then \operatorname F \leq , \operatorname F, \leq , F, \leq p. For the converse, assume \operatorname F \leq p and fix x \in X. Let r = , F(x), and pick any \theta \in \R such that F(x) = r e^; it remains to show r \leq p(x). Homogeneity of F implies F\left(e^ x\right) = r is real so that \operatorname F\left(e^ x\right) = F\left(e^ x\right). By assumption, \operatorname F \leq p and p\left(e^ x\right) = p(x), so that r = \operatorname F\left(e^ x\right) \leq p\left(e^ x\right) = p(x), as desired. \blacksquare
, F, \,\leq\, p \quad \text \quad \operatorname F \,\leq\, p.
Stated in simpler language, a linear functional is
dominated by a seminorm
p if and only if its
real part is dominated above by
p.
The proof above shows that when
p is a seminorm then there is a one-to-one correspondence between dominated linear extensions of
f : M \to \Complex and dominated real-linear extensions of
\operatorname f : M \to \R; the proof even gives a formula for explicitly constructing a linear extension of
f from any given real-linear extension of its real part.
Continuity
A linear functional
F on a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
if and only if this is true of its real part
\operatorname F; if the domain is a normed space then
\, F\, = \, \operatorname F\, (where one side is infinite if and only if the other side is infinite).
Assume
X is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and
p : X \to \R is
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
.
If
p is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
sublinear function that dominates a linear functional
F then
F is necessarily continuous. Moreover, a linear functional
F is continuous if and only if its
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, F, (which is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
that dominates
F) is continuous. In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.
Proof
The
Hahn–Banach theorem for real vector spaces ultimately follows from Helly's initial result for the special case where the linear functional is extended from
M to a larger vector space in which
M has
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
1.
This lemma remains true if
p : X \to \R is merely a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
instead of a sublinear function.
Assume that
p is convex, which means that
p(t y + (1 - t) z) \leq t p(y) + (1 - t) p(z) for all
0 \leq t \leq 1 and
y, z \in X. Let
M, f : M \to \R, and
x \in X \setminus M be as in
the lemma's statement. Given any
m, n \in M and any positive real
r, s > 0, the positive real numbers
t := \tfrac and
\tfrac = 1 - t sum to
1 so that the convexity of
p on
X guarantees
\begin
p\left(\tfrac m + \tfrac n\right)
~&=~ p\big(\tfrac (m - r x) &&+ \tfrac (n + s x)\big) && \\
&\leq~ \tfrac \; p(m - r x) &&+ \tfrac \; p(n + s x) && \\
\end
and hence
\begin
s f(m) + r f(n)
~&=~ (r + s) \; f\left(\tfrac m + \tfrac n\right) && \qquad \text f \\
&\leq~ (r + s) \; p\left(\tfrac m + \tfrac n\right) && \qquad f \leq p \text M \\
&\leq~ s p(m - r x) + r p(n + s x) \\
\end
thus proving that
- s p(m - r x) + s f(m) ~\leq~ r p(n + s x) - r f(n), which after multiplying both sides by
\tfrac becomes
\tfrac p(m - r x) + f(m)~\leq~ \tfrac (n + s x) - f(n)
This implies that the values defined by
a = \sup_ \tfrac p(m - r x) + f(m)\qquad \text \qquad c = \inf_ \tfrac (n + s x) - f(n)/math>
are real numbers that satisfy a \leq c. As in the above proof of the one–dimensional dominated extension theorem above, for any real b \in \R define F_b : M \oplus \R x \to \R by F_b(m + r x) = f(m) + r b.
It can be verified that if a \leq b \leq c then F_b \leq p where r b \leq p(m + r x) - f(m) follows from b \leq c when r > 0 (respectively, follows from a \leq b when r < 0).
\blacksquare
The lemma above is the key step in deducing the dominated extension theorem from Zorn's lemma.
When M has countable codimension, then using induction and the lemma completes the proof of the Hahn–Banach theorem. The standard proof of the general case uses Zorn's lemma although the strictly weaker ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
(which is equivalent to the compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally no ...
and to the Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filt ...
) may be used instead. Hahn-Banach can also be proved using Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
for compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s (which is also equivalent to the ultrafilter lemma)
The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.
In locally convex spaces
In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question
In classical rhetoric and logic, begging the question or assuming the conclusion (Latin: ') is an informal fallacy that occurs when an argument's premises assume the truth of the conclusion, instead of supporting it.
For example:
* "Green is t ...
. However, in locally convex spaces
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological spa ...
, any continuous functional is already bounded by the norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
, which is sublinear. One thus hasIn category-theoretic terms, the field \mathbf is an injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. ...
in the category of locally convex vector spaces.
Geometric Hahn–Banach (the Hahn–Banach separation theorems)
The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: \, and \. This sort of argument appears widely in convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...
, optimization theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, and economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.
When the convex sets have additional properties, such as being open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
or compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
for example, then the conclusion can be substantially strengthened:
Then following important corollary is known as the Geometric Hahn–Banach theorem or Mazur's theorem. It follows from the first bullet above and the convexity of M.
Mazur's theorem clarifies that vector subspaces (even those that are not closed) can be characterized by linear functionals.
Supporting hyperplanes
Since points are trivially convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, geometric Hahn-Banach implies that functionals can detect the boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of a set. In particular, let X be a real topological vector space and A \subseteq X be convex with \operatorname A \neq \varnothing. If a_0 \in A \setminus \operatorname A then there is a functional that is vanishing at a_0, but supported on the interior of A.
Call a normed space X smooth if at each point x in its unit ball there exists a unique closed hyperplane to the unit ball at x. Köthe showed in 1983 that a normed space is smooth at a point x if and only if the norm is Gateaux differentiable at that point.
Balanced or disked neighborhoods
Let U be a convex balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
neighborhood of the origin in a locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
topological vector space X and suppose x \in X is not an element of U. Then there exists a continuous linear functional f on X such that
\sup , f(U), \leq , f(x), .
Applications
The Hahn–Banach theorem is the first sign of an important philosophy in functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
: to understand a space, one should understand its continuous functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
s.
For example, linear subspaces are characterized by functionals: if is a normed vector space with linear subspace (not necessarily closed) and if z is an element of not in the closure of , then there exists a continuous linear map f : X \to \mathbf with f(m) = 0 for all m \in M, f(z) = 1, and \, f\, = \operatorname(z, M)^. (To see this, note that \operatorname(\cdot, M) is a sublinear function.) Moreover, if z is an element of , then there exists a continuous linear map f : X \to \mathbf such that f(z) = \, z\, and \, f\, \leq 1. This implies that the natural injection J from a normed space into its double dual
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
V^ is isometric.
That last result also suggests that the Hahn–Banach theorem can often be used to locate a "nicer" topology in which to work. For example, many results in functional analysis assume that a space is Hausdorff or locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
. However, suppose is a topological vector space, not necessarily Hausdorff or locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
, but with a nonempty, proper, convex, open set . Then geometric Hahn-Banach implies that there is a hyperplane separating from any other point. In particular, there must exist a nonzero functional on — that is, the continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
X^* is non-trivial. Considering with the weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
induced by X^*, then becomes locally convex; by the second bullet of geometric Hahn-Banach, the weak topology on this new space separates points
''Separates'' is the second album by English punk rock band 999, released in 1978. ''Separates'' was released in the United States under the title ''High Energy Plan'', with a different cover and slightly altered track listing; on ''High Energ ...
.
Thus with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.
Partial differential equations
The Hahn–Banach theorem is often useful when one wishes to apply the method of a priori estimate
In the theory of partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The functi ...
s. Suppose that we wish to solve the linear differential equation P u = f for u, with f given in some Banach space . If we have control on the size of u in terms of \, f\, _X and we can think of u as a bounded linear functional on some suitable space of test functions g, then we can view f as a linear functional by adjunction: (f, g) = (u, P^*g). At first, this functional is only defined on the image of P, but using the Hahn–Banach theorem, we can try to extend it to the entire codomain . The resulting functional is often defined to be a weak solution to the equation.
Characterizing reflexive Banach spaces
Example from Fredholm theory
To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem.
The above result may be used to show that every closed vector subspace of \R^ is complemented because any such space is either finite dimensional or else TVS–isomorphic to \R^.
Generalizations
General template
There are now many other versions of the Hahn–Banach theorem. The general template for the various versions of the Hahn–Banach theorem presented in this article is as follows:
:p : X \to \R is a sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
(possibly a seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
) on a vector space X, M is a vector subspace of X (possibly closed), and f is a linear functional on M satisfying , f, \leq p on M (and possibly some other conditions). One then concludes that there exists a linear extension F of f to X such that , F, \leq p on X (possibly with additional properties).
For seminorms
Geometric separation
Maximal dominated linear extension
If S = \ is a singleton set (where s \in X is some vector) and if F : X \to \R is such a maximal dominated linear extension of f : M \to \R, then F(s) = \inf_ (s) + p(s - m)
Vector valued Hahn–Banach
For nonlinear functions
The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem.
The following theorem characterizes when scalar function on X (not necessarily linear) has a continuous linear extension to all of X.
Converse
Let be a topological vector space. A vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on , and we say that has the Hahn–Banach extension property (HBEP) if every vector subspace of has the extension property.
The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
s there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex. On the other hand, a vector space of uncountable dimension, endowed with the finest vector topology
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
, then this is a topological vector spaces with the Hahn-Banach extension property that is neither locally convex nor metrizable.
A vector subspace of a TVS has the separation property if for every element of such that x \not\in M, there exists a continuous linear functional f on such that f(x) \neq 0 and f(m) = 0 for all m \in M. Clearly, the continuous dual space of a TVS separates points on if and only if \, has the separation property. In 1992, Kakol proved that any infinite dimensional vector space , there exist TVS-topologies on that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on . However, if is a TVS then vector subspace of has the extension property if and only if vector subspace of has the separation property.
Relation to axiom of choice and other theorems
The proof of the Hahn–Banach theorem for real vector spaces (HB) commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(ZF) is equivalent to the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
(AC). It was discovered by Łoś and Ryll-Nardzewski and independently by Luxemburg
Luxembourg ( ; lb, Lëtzebuerg ; french: link=no, Luxembourg; german: link=no, Luxemburg), officially the Grand Duchy of Luxembourg, ; french: link=no, Grand-Duché de Luxembourg ; german: link=no, Großherzogtum Luxemburg is a small land ...
that HB can be proved using the ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
(UL), which is equivalent (under ZF) to the Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that Ideal (order theory), ideals in a Boolean algebra (structure), Boolean algebra can be extended to Ideal (order theory)#Prime ideals , prime ideals. A variation of this statement for Filt ...
(BPI). BPI is strictly weaker than the axiom of choice and it was later shown that HB is strictly weaker than BPI.
The ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
is equivalent (under ZF) to the Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.
A common proo ...
, which is another foundational theorem in functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. Although the Banach–Alaoglu theorem implies HB, it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than HB).
However, HB is equivalent to a certain weakened version of the Banach–Alaoglu theorem for normed spaces.
The Hahn–Banach theorem is also equivalent to the following statement:
:(∗): On every Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
there exists a "probability charge", that is: a non-constant finitely additive map from B into , 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
(BPI is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.)
In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set. Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
.
For separable Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
A precurs ...
that takes a form of Kőnig's lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computab ...
restricted to binary trees as an axiom. In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
.[Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, , ]
See also
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Notes
Proofs
References
Bibliography
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* Reed, Michael and Simon, Barry, ''Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis,'' Section III.3. Academic Press, San Diego, 1980. .
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* Tao, Terence
The Hahn–Banach theorem, Menger's theorem, and Helly's theorem
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* Wittstock, Gerd, Ein operatorwertiger Hahn-Banach Satz
J. of Functional Analysis 40 (1981), 127–150
* Zeidler, Eberhard, ''Applied Functional Analysis: main principles and their applications'', Springer, 1995.
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{{DEFAULTSORT:Hahn-Banach theorem
Articles containing proofs
Linear algebra
Theorems in functional analysis
Topological vector spaces