Hahn–Banach theorem

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 ${\displaystyle {C}{\left[a,b\right]}}$ of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly,[1] and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz.[2]

The first Hahn-Banach theorem was proved by Eduard Helly in 1921 who showed that certain linear functionals defined on a subspace of a certain type of normed space (${\displaystyle \mathbb {C} ^{\mathbb {N} }}$) had an extension of the same norm.[3] Helly did this through the technique of first proving that a one-dimensional extension exists (where the linear functional has its domain extended by one dimension) and then using induction. In 1927, Hahn defined general Banach spaces and used Helly's technique to prove a norm preserving version of Hahn-Banach theorem for Banach spaces (where a bounded linear functional on a subspace has a bounded linear extension of the same norm to the whole space).[3] In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses sublinear functionals.[3] Whereas Helly's proof used mathematical induction, Hahn and Banach both used transfinite induction.[3]

The Hahn-Banach theorem arose from attempts to solve infinite systems of linear equations. This is needed to solve problems such as the moment problem, whereby given all the potential moments of a function one must determine if a function having these moments exists and if so then find it. Another such problem is the Fourier cosine series problem, whereby given all the potential Fourier cosine coefficients one must determine if a function having those coefficients exists and if so then find it.[3] Riesz and Helly solved the problem for certain classes of spaces (such as L^p([0, 1]) and C([a, b])) where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals.[3] In effect, they needed to solve the following problem:[3]

(The vector problem) Given a collection ${\displaystyle \left(f_{i}\right)_{i\in I}}$ of bounded linear functionals on a normed space X and a collection of scalars ${\displaystyle \left(c_{i}\right)_{i\in I}}$, determine if there is an x ∈ X such that f_i(x) = c_i for all i ∈ I.

To solve this, if X is reflexive then it suffices to solve the following dual problem:[3]

(The functional problem) Given a collection ${\displaystyle \left(x_{i}\right)_{i\in I}}$ of vectors in a normed space X and a collection of scalars ${\displaystyle \left(c_{i}\right)_{i\in I}}$, determine if there is a bounded linear functional f on X such that f(x_i) = c_i for all i ∈ I.[3]

Riesz went on to define L^p([0, 1]) (1 < p < ∞) in 1910 and the l^p spaces in 1913.[3] While investing these spaces he proved a special case of the Hahn-Banach theorem. Hally also proved a special case of the Hahn-Banach theorem in 1912.[3] 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.[3]

Theorem (The functional problem):[3] Let X be a real or complex normed space, I a non-empty set, ${\displaystyle \left(c_{i}\right)_{i\in I}}$ a family of scalars, and ${\displaystyle \left(x_{i}\right)_{i\in I}}$ a family of vectors. Then there exists a continuous linear functional f on X such that f(x_i) = c_i for all i ∈ I if and only if there exists a K > 0 such that for any choice of scalars ${\displaystyle \left(s_{i}\right)_{i\in I}}$ where all but finitely many s_i are 0, we necessarily have

${\displaystyle \left|\sum _{i\in I}s_{i}c_{i}\right|\leq K\left\|\sum _{i\in I}s_{i}x_{i}\right\|}$.

One can use the above theorem to deduce the Hahn-Banach theorem.[3] If X is reflexive, then this theorem solves the vector problem.

The most general formulation of the theorem needs some preparation. Given a real vector space X, a function p : X → R is called sublinear if

• Positive homogeneity: p(γx) = γ p(x) for all γ ∈ R₊, x ∈ X,
• Subadditivity: p(x + y) ≤ p(x) + p(y) for all x, y ∈ X.

Every seminorm on X (in particular, every norm on X) and every linear functional on X is sublinear. Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

If f is a linear functional on a topological vector space (TVS) X (e.g. a normed space) and if p is a continuous sublinear functional on X then |f| ≤ p implies that f is continuous. If f is a real-valued linear functional and p is a seminorm (not just a sublinear functional) then f ≤ p implies that |f| ≤ p.[3] If f(x) = R(x) + i I(x) is a complex-valued linear functional on a complex vector space X then I(x) = - R(ix) for all x ∈ X, so that I is completely determined by f's real part R, which we will also denote by Re f, and this in turn implies that f is entirely determined by R.[3] In particular, it follows that f is bounded (resp. continuous) if and only if R is bounded (resp. continuous).

If S is a subset of X and if f : S → R is a function then we say that p dominates f on S if f(s) ≤ p(s) for all s ∈ S. We call a function F : X → R and extension of f to X if F(s) = f(s) for all s ∈ S; if in addition F is a linear map then we call F a linear extension of f.

The general template for the various versions of the Hahn-Banach theorem presented in this article is as follows:

X is a vector space, p is a sublinear functional on X, M is a vector subspace of X, and f is a linear functional on M satisfying (for instance) |f| ≤ p on M (and possibly some other conditions). One then concludes (possibly among other things) that there exists a linear extension F of f to X such that |F| ≤ p on X.

Sometimes X is assumed to have additional structure, such as a topology or a norm, but many of the statements are purely algebraic. We may apply a purely algebraic version of this theorem to the case where X is a topological vector space (TVS) (e.g. a normed space) as follows: by choosing p to be continuous, the inequality |F| ≤ p allows us to conclude that F is necessarily continuous. (As a side note, if p is continuous then the assumption "|f| ≤ p on M" also allows us to conclude that f is continuous).

Some of the statements are given only for real vector spaces or only real-valued linear functionals while others are given for real or complex vector spaces. One may apply a result that applies only to real-valued linear functionals to the complex case by recalling that a complex-valued linear functional c(x) = R(x) + i I(x) is continuous if and only if its real part, R, is continuous and that furthermore, the real part R completely determines the imaginary part I and thus completely determines c. The sublinear functional p is always real-valued (although it could possibly take on negative values if it is not assumed to be positive). If X is assumed to be a real (resp. complex) vector space then all linear functionals are assumed to be real-valued (resp. complex-valued). If the linear functional f is real-valued then you'll often see the condition f ≤ p whereas if f is complex-valued then you're more likely to see |f| ≤ p or Re f ≤ p.

Sometimes a functional is assumed to be bounded and other times it is assumed to be continuous. If X a is pseudometrizable TVS (e.g. a metrizable TVS or a normed space) then a linear map from X into any other TVS is continuous if and only if it is a bounded map (i.e. it maps bounded subsets of X to bounded subsets of the codomain). In particular, a linear functional on a normed or Banach space is continuous if and only if it is bounded.

The following lemma is fundamental to proving the general Hahn-Banach theorem and its basic prove first appeared in a 1912 paper by Helly where it was proved for the space C([a, b]).[3]

Lemma (One-dimensional dominated extension theorem):[3] Let X be a real vector space, p : X → ℝ a sublinear function, f : M → ℝ a linear functional on a proper vector subspace M ⊆ X such that f ≤ p on M (i.e. f(m) ≤ p(m) for all m ∈ M), and x ∈ X an vector not in M. There exists a linear extension F : M ⊕ ℝx → ℝ of f to M ⊕ ℝx = span { M, x } such that F ≤ p on M ⊕ ℝx.

To prove this lemma, one first shows that for all m, n ∈ M, -p(- x - n) - f(n) ≤ p(m + x) - f(m) which allows us to define:

a = ${\displaystyle \sup _{n\in M}}$ [ -p(- x - n) - f(n)], and b = ${\displaystyle \inf _{m\in M}}$ [p(m + x) - f(m)]

from which we conclude "the decisive inequality"[3] that for any c ∈ [a, b],

-p(- x - n) - f(n) ≤ c ≤ p(m + x) - f(m).

For any m + rx ∈ M ⊕ ℝx, one then defines F(m + rx) := f(m) + rc, which gives us the desired extension.

Hahn–Banach dominated extension theorem:[3](Rudin 1991, Th. 3.2). If p : X → ℝ is a sublinear function, and f : M → R is a linear functional on a linear subspace M ⊆ X which is dominated by p on M, then there exists a linear extension F : X → ℝ of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that

${\displaystyle F(x)=f(x)\qquad \forall x\in M,}$

${\displaystyle F(x)\leq p(x)\qquad \forall x\in X.}$

Hahn–Banach theorem (alternative version). Set K = ℝ or C and let X be a K-vector space with a seminorm p : X → ℝ. If f : M → K is a K-linear functional on a K-linear subspace M of X which is dominated by p on M in absolute value,

${\displaystyle |f(x)|\leq p(x)\qquad \forall x\in M}$

then there exists a linear extension F : X → K of f to the whole space X, i.e., there exists a K-linear functional F such that

${\displaystyle F(x)=f(x)\;\qquad \forall x\in M,}$

${\displaystyle |F(x)|\leq p(x)\qquad \forall x\in X.}$

In the complex case of the alternate version, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ M, we have ix ∈ M and f(ix) = if(x).

The extension F is in general not uniquely specified by f and the proof gives no explicit method as to how to find F. The usual proof for the case of an infinite dimensional space X uses Zorn's lemma or, equivalently, the axiom of choice. It is now known (see below) that the ultrafilter lemma, which is slightly weaker than the axiom of choice, is actually strong enough.

It is possible to relax slightly the subadditivity condition on p, requiring only that (Reed and Simon, 1980):

${\displaystyle p(ax+by)\leq |a|\,p(x)+|b|\,p(y),\qquad x,y\in X,\quad |a|+|b|\leq 1.}$

It is further possible to relax the positive homogeneity and the subadditivity conditions on p, requiring only that p is convex (Schechter, 1996).

This reveals the intimate connection between the Hahn–Banach theorem and convexity.

The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file.[4]

Theorem:[3] If p is a sublinear functional on a real vector space X then there exists a linear functional f on X such that f ≤ p on X.

Theorem:[3] Let p be a sublinear functional on a real vector space X and let z ∈ X. Then there exists a linear functional f on X such that

1. f(z) = p(z);
2. -p(-x) ≤ f(x) ≤ p(x) for all x ∈ X;
3. if p is a seminorm then | f | ≤ p.

If X is a TVS and p is continuous at 0, then f is continuous.

The theorem has several important consequences, some of which are also sometimes called "Hahn–Banach theorem":

• (Continuous extensions on locally convex spaces): Let X be a real or complex locally convex topological vector space (TVS), M a vector subspace of X, and f a continuous linear functional on M. There f has a continuous linear extension to all of X.[3]

For normed spaces we have the following results:

• If X is a normed vector space with linear subspace M (not necessarily closed) and if f : M → K is continuous and linear, then there exists an extension F : X → K of f which is also continuous and linear and which has the same operator norm as f (see Banach space for a discussion of the norm of a linear map). In other words, in the category of normed vector spaces, the space K is an injective object.
• If X is a normed vector space with linear subspace M (not necessarily closed) and if z is an element of X not in the closure of M, then there exists a continuous linear map f : X → K with f(x) = 0 for all x in M, f(z) = 1, and ||f|| = dist(z, M)⁻¹.
• In particular, if X is a normed vector space and z is an element of X, then there exists a continuous linear map f : X → K such that f(z) = ||z|| and ||f|| ≤ 1. This implies that the natural injection J from a normed space X into its double dual V′′ is isometric.
• If X is a normed space over the real or complex numbers, M is a closed proper vector subspace of X, and f is a continuous linear functional on M, then there exists a continuous linear extension F of f to X such that ${\displaystyle \|F\|>\|f\|}$.[3]

Hahn–Banach separation theorems are the geometrical versions of the Hahn–Banach Theorem.[5][6] They have numerous uses in convex geometry,[7] optimization theory, and economics. The separation theorem is derived from the original form of the theorem.

Let X be a real vector space, A and B non-empty subsets of X, f ≠ 0 a real linear functional on X, s a scalar, and let ${\displaystyle H=f^{-1}(s)}$. We also define the lower (resp. upper) half space to be { x ∈ X : f(x) ≤ s } (resp. { x ∈ X : f(x) ≥ s }). We define the strict lower (resp. strict upper) half space to be { x ∈ X : f(x) < s } (resp. { x ∈ X : f(x) > s }).

We say that we say that H (or f) separates A and B if sup f(A) ≤ s ≤ inf f(B) or equivalently, if f(a) ≤ s ≤ f(b) for all a ∈ A and b ∈ B. The separation is:

• proper if ${\displaystyle A\cup B\not \subseteq H}$;[5]
• strict if A ∩ H = ∅ and B ∩ H = ∅ or equivalently, if f(a) < s < f(b) for all a ∈ A and b ∈ B[3] (note that some authors define "strict" to mean that A ∩ H = ∅ or B ∩ H = ∅);[5]
• strong and we say that A and B are strongly separated by H if there exists an r > 0 such that f(a) ≤ s - r < s + r ≤ f(b) for all a ∈ A and b ∈ B.[3]

Note that A and B are separated (resp. strictly separated, strongly separated) if and only if the same is true of { 0 } and B - A.[3] If A and B are convex then they are strongly separated by a hyperplane if and only if there exists an absorbing convex U such that (A + U) ∩ B = ∅.[3] We say that A and B are united if they cannot be properly separated.[5]

If a₀ ∈ A and H separates A and { a₀ } then H is called a supporting hyperplane of A at a₀, a₀ is called a support point of A, and f is called a support functional.[5] If A is convex and a₀ ∈ A, then we call a₀ a smooth point of A if there exists a unique hyperplane H such that a₀ ∈ A ∩ H.[3] We 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.[3] 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.[3]

Hahn-Banach sandwich theorem:[3] Let S be any subset of a real vector space X, let p be a sublinear functional on X, and let f : S → ℝ be any map. If there exist positive numbers a and b such that for all x, y ∈ S,

${\displaystyle 0\geq \inf _{s\in S}\left[p\left(s-ax-by\right)-f(s)-af(x)-bf(y)\right]}$

then there exists a linear functional F on X such that F ≤ p on X and f ≤ F on S.

One form of Hahn–Banach theorem is known as the Geometric Hahn–Banach theorem, or Mazur's theorem.[9]

Theorem. Let K be a convex set having a nonempty interior in a real normed linear vector space X. Suppose X is a linear variety in X containing no interior points of K. Then there is a closed hyperplane in X containing X but containing no interior points of K; i.e., there is an element x* ∈ X* and a constant c such that ${\displaystyle \langle v,x^{*}\rangle =c}$ for all v ∈ X and ${\displaystyle \langle k,x^{*}\rangle <c}$ for all k ∈ int(K).

This can be generalized to an arbitrary topological vector space, which need not be locally convex or even Hausdorff, as:[10]

Theorem. Let M be a vector subspace of the topological vector space X. Suppose K is a non-empty convex open subset of X with K ∩ M = ∅. Then there is a closed hyperplane N in X containing M with K ∩ N = ∅.

The following theorem of Mazur-Orlicz (1953) is equivalent to the Hahn-Banach theorem.

Mazur-Orlicz theorem:[3] Let T be any set, r : T → ℝ be any real-valued map, X be a real or complex vector space, v : T → X be any map, and p be a sublinear functional on X. Then the following are equivalent:

1. there exists a real-valued linear functional F on X such that F ≤ p on X and r ≤ F ∘ v on T;
2. for any positive integer n, any sequence s₁, ..., s_n of non-negative real numbers, and any sequence t₁, ..., t_n of elements of T, ${\displaystyle \sum _{i=1}^{n}s_{i}r\left(t_{i}\right)\leq p\left(\sum _{i=1}^{n}s_{i}v\left(t_{i}\right)\right)}$

Due to its importance, the Hahn-Banach theorem has been generalized many times.

Theorem (Andenaes, 1970):[3] Let M be a vector subspace of a real vector space X, p be a sublinear functional on X, f be a linear functional on M such that f ≤ p on M, and let S be any subset of X. Then there exists a linear functional F on X that extends f, satisfies F ≤ p on X, and is (pointwise) maximal in the following sense: if G is a linear functional on X ending f and satisfying G ≤ p on X, then G ≥ F implies that G = F on S.

Theorem:[3] Let X and Y be vector spaces over the same field, M be a vector subspace of X, and f : M → Y be a linear map. Then there exists a linear map F : X → Y that extends f.

Theorem:[3] Let M be a vector subspace of a real or complex vector space X, let D be an absorbing disk in X, and let f be a linear functional on M such that |f| ≤ 1 on M ∩ D. Then there exists a linear functional F on X extending f such that |F| ≤ 1 on D.

As mentioned earlier, the axiom of choice implies the Hahn–Banach theorem. The converse is not true. One way to see that is by noting that the ultrafilter lemma (or equivalently, the Boolean prime ideal theorem), which is strictly weaker than the axiom of choice, can be used to show the Hahn–Banach theorem, although the converse is not the case.

The Hahn–Banach theorem is equivalent to the following:[11]

(∗): On every Boolean algebra B there exists a "probability charge", that is: a nonconstant finitely additive map from B into [0, 1].

(The Boolean prime ideal theorem is easily seen to be equivalent to the statement that there are always nonconstant probability charges which take only the values 0 and 1.)

In Zermelo–Fraenkel set theory, one can show that the Hahn–Banach theorem is enough to derive the existence of a non-Lebesgue measurable set.[12] Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.[13]

For separable Banach spaces, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL₀, a weak subsystem of second-order arithmetic 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.[14][15]

• Fichera's existence principle
• M. Riesz extension theorem
• Separating axis theorem
• Farkas' lemma

ISBN 0-12-622760-8.

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