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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Hahn–Banach theorem is a central result that allows the extension of
bounded linear functionals defined on a
vector subspace of some
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
to the whole space. The theorem also shows that there are sufficient
continuous linear functionals defined on every
normed vector space in order to study the
dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the
hyperplane separation theorem, and has numerous uses in
convex geometry.
History
The theorem is named for the mathematicians
Hans Hahn and
Stefan Banach, who proved it independently in the late 1920s.
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 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 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. 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 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.
For this reason, 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 distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
:
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 semino ...
is a convex function.
On the other hand, if
p : X \to \R is convex with
p(0) \geq 0, then the function defined by
p_0(x) \;\stackrel\; \inf_ \frac is
positively homogeneous
(because for all
x and
r>0 one has
p_0(rx)=\inf_ \frac =r\inf_ \frac = r\inf_ \frac=rp_0(x)), hence, being convex,
it is sublinear. It is also bounded above by
p_0 \leq p, and satisfies
F \leq p_0 for every linear functional
F \leq p. So the extension of the Hahn–Banach theorem to convex functionals does not have 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 is a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
and both are symmetric
balanced sublinear functions. A sublinear function is a seminorm if and only if it is a
balanced function. 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 \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 and
balanced function 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 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 like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
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 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 semino ...
, 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 \; \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 on
X then their
dual norm
In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
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 (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 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 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 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 semino ...
.
If
p is a
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, F, (which is a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
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 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 distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
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
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
.
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
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
although the strictly weaker ultrafilter lemma (which is equivalent to the compactness theorem and to the Boolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
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.
Continuous extension theorem
The Hahn–Banach theorem can be used to guarantee the existence of continuous linear extensions of continuous linear functionals.
In category-theoretic terms, the underlying field of the vector space is an injective object in the category of locally convex vector spaces.
On a normed (or seminormed) space, a linear extension F of a bounded linear functional f is said to be if it has the same dual norm
In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
as the original functional: \, F\, = \, f\, .
Because of this terminology, the second part of the above theorem is sometimes referred to as the " norm-preserving" version of the Hahn–Banach theorem. Explicitly:
Proof of the continuous extension theorem
The following observations allow the continuous extension theorem to be deduced from the Hahn–Banach theorem.
The absolute value of a linear functional is always a seminorm. 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 ...
X is continuous if and only if its absolute value , F, is continuous, which happens if and only if there exists a continuous seminorm p on X such that , F, \leq p on the domain of F.
If X is a locally convex space then this statement remains true when the linear functional F is defined on a vector subspace of X.
Proof for normed spaces
A linear functional f on a normed space is continuous if and only if it is bounded, which means that its dual norm
In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
\, f\, = \sup \
is finite, in which case , f(m), \leq \, f\, \, m\, holds for every point m in its domain.
Moreover, if c \geq 0 is such that , f(m), \leq c \, m\, for all m in the functional's domain, then necessarily \, f\, \leq c.
If F is a linear extension of a linear functional f then their dual norms always satisfy \, f\, \leq \, F\,
so that equality \, f\, = \, F\, is equivalent to \, F\, \leq \, f\, , which holds if and only if , F(x), \leq \, f\, \, x\, for every point x in the extension's domain.
This can be restated in terms of the function \, f\, \, \, \cdot\, : X \to \Reals defined by x \mapsto \, f\, \, \, x\, , which is always a seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
:[Like every non-negative scalar multiple of a norm, this seminorm \, f\, \, \, \cdot\, (the product of the non-negative real number \, f\, with the norm \, \cdot\, ) is a norm when \, f\, is positive, although this fact is not needed for the proof.]
:A linear extension of a bounded linear functional f is norm-preserving if and only if the extension is dominated by the seminorm \, f\, \, \, \cdot\, .
Applying the Hahn–Banach theorem to f with this seminorm \, f\, \, \, \cdot\, thus produces a dominated linear extension whose norm is (necessarily) equal to that of f, which proves the theorem:
Non-locally convex spaces
The continuous extension theorem might fail if the 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 ...
(TVS) X is not locally convex. For example, for 0 < p < 1, the Lebesgue space L^p( , 1 is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself L^p( , 1 and the empty set) and the only continuous linear functional on L^p( , 1 is the constant 0 function . Since L^p( , 1 is Hausdorff, every finite-dimensional vector subspace M \subseteq L^p( , 1 is linearly homeomorphic to Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\Reals^ or \Complex^ (by F. Riesz's theorem) and so every non-zero linear functional f on M is continuous but none has a continuous linear extension to all of L^p( , 1.
However, it is possible for a TVS X to not be locally convex but nevertheless have enough continuous linear functionals that its continuous dual space X^* separates points; for such a TVS, a continuous linear functional defined on a vector subspace have a continuous linear extension to the whole space.
If the TVS X is not locally convex then there might not exist any continuous seminorm p : X \to \R (not just on M) that dominates f, in which case the Hahn–Banach theorem can not be applied as it was in the above proof of the continuous extension theorem.
However, the proof's argument can be generalized to give a characterization of when a continuous linear functional has a continuous linear extension: If X is any TVS (not necessarily locally convex), then a continuous linear functional f defined on a vector subspace M has a continuous linear extension F to all of X if and only if there exists some continuous seminorm p on X that dominates f. Specifically, if given a continuous linear extension F then p := , F, is a continuous seminorm on X that dominates f; and conversely, if given a continuous seminorm p : X \to \Reals on X that dominates f then any dominated linear extension of f to X (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension.
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, optimization theory, and economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.
They are generalizations of the hyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space \R^n can be separated by some , which is a fiber
Fiber (spelled fibre in British English; from ) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often inco ...
( level set) of the form f^(s) = \ where f \neq 0 is a non-zero linear functional and s is a scalar.
When the convex sets have additional properties, such as being open or compact 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 (also known as Ascoli–Mazur 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, geometric Hahn–Banach implies that functionals can detect the boundary 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 neighborhood of the origin in a locally convex 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
: to understand a space, one should understand its continuous functionals.
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 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. However, suppose is a topological vector space, not necessarily Hausdorff or locally convex, 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 X^* is non-trivial. Considering with the weak topology induced by X^*, then becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space separates points.
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 estimates. 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 semino ...
(possibly a seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
) 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
So for example, suppose that f is a bounded linear functional defined on a vector subspace M of a normed space X, so its the operator norm \, f\, is a non-negative real number.
Then the linear functional's 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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
p := , f, is a seminorm on M and the map q : X \to \Reals defined by q(x) = \, f\, \, \, x\, is a seminorm on X that satisfies p \leq q\big\vert_M on M.
The Hahn–Banach theorem for seminorms guarantees the existence of a seminorm P : X \to \Reals that is equal to , f, on M (since P\big\vert_M = p = , f, ) and is bounded above by P(x) \leq \, f\, \, \, x\, everywhere on X (since P \leq q).
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
Invariant Hahn–Banach
A set \Gamma of maps X \to X is (with respect to function composition \,\circ\,) if F \circ G = G \circ F for all F, G \in \Gamma.
Say that a function f defined on a subset M of X is if L(M) \subseteq M and f \circ L = f on M for every L \in \Gamma.
This theorem may be summarized:
:Every \Gamma-invariant continuous linear functional defined on a vector subspace of a normed space X has a \Gamma-invariant Hahn–Banach extension to all of X.
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 spaces 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, 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
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
, which in the axiomatic framework of Zermelo–Fraenkel set theory (ZF) is equivalent to the axiom of choice (AC). It was discovered by Łoś and Ryll-Nardzewski and independently by Luxemburg that HB can be proved using the ultrafilter lemma (UL), which is equivalent (under ZF) to the Boolean prime ideal theorem (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 is equivalent (under ZF) to the Banach–Alaoglu theorem, 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. 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 there exists a "probability charge", that is: a non-constant finitely additive map from B into , 1
(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.
For separable Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
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 of mathematics, foundation for much, but not all, ...
that takes a form of Kőnig's lemma 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.[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
Linear functionals
Theorems in functional analysis
Topological vector spaces