Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis (a field of mathematics). It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space 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, 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 ${\displaystyle {C}{\left[a,b\right]}}$ 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 functions.[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 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.[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[3] (The functional problem) — Let X be a real or complex normed space, I a non-empty set, (c_i)_{i ∈ I} a family of scalars, and (x_i)_{i ∈ I} a family of vectors in X.

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 (s_i)_{i ∈ 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.

Basic definitions

Algebraic definitions

We will assume that X is a vector space over the field 𝔽, where 𝔽 is called the underlying (scalar) field of X and we assume that 𝔽 is either the real numbers ℝ or complex numbers ℂ. A scalar is any element of X 's underlying scalar field 𝔽.

We now define some important properties that function f from X into another vector space may have:

1. Nonnegative/Positive: f is real-valued and never takes on a negative value.
2. Positive definiteness/Separates points: if x ∈ X satisfies f (x) = 0 then x = 0.
3. Additivity: f (x + y) = f (x) + f (y) for all x, y ∈ X.
4. Subadditivity/Triangle inequality: f (x+y) ≤ f (x) + f (y) for all x, y ∈ X (this assume that f is real-valued).
5. Nonnegative homogeneity: f (r x) = r f (x) for any non-negative real r ≥ 0 and any x ∈ X.
6. Positive homogeneity: This is usually defined to mean "nonnegative homogeneity" but it is also frequently defined to instead mean "strict positive homogeneity": f (r x) = r f (x) for all x ∈ X and all positive real r > 0. We now show that nonnegative homogeneity is equivalent to strict positive homogeneity.
   • Assume that f is strictly positively homogeneous. Then f (0) = f (2 ⋅ 0) = 2 f (0) implies f (0) = 0. Let r := 0 and note that for all x ∈ X we have f (r x) = f (0 x) = f (0) = 0 = 0 f (x) = r f (x). Thus f is nonnegative homogeneous.
   • Every non-negative ℝ-valued function on X that is nonnegative homogeneous is equal to the Minkowski functional of some subset of X (see the article Minkowski functional for details).
7. Real homogeneity: f (r x) = r f (x) for all x ∈ X and all real r.
   • It can be shown that f is real homogeneous if and only if f (rx) = r f (x) for all x ∈ X and all non-zero real r ≠ 0.
   • Real homogeneity is equivalent to f being nonnegative homogeneous and commuting with negation: - f (x) = f (- x) for all x ∈ X.
   • Non-trivial seminorms are examples of maps that are nonnegative homogeneous but not real homogeneous.
8. Homogeneity: f (s x) = s f (x) or all x ∈ X and all scalars s ∈ 𝔽 (where recall that 𝔽 is the underlying scalar field of X).
   • It is emphasized the definition of homogeneity depends on the underlying scalar field 𝔽.
     • In particular, the only homogeneous real-valued function on a complex vector space is the trivial map.
   • If 𝔽 = ℝ then homogeneity is equivalent to real homogeneity.
9. Absolute homogeneity: f (s x) = |s| f (x) for all x ∈ X and all scalars s ∈ 𝔽.
   • If a function is absolutely homogeneous then it is nonnegative homogeneous and satisfies f (x) = f (- x) for all x ∈ X. If 𝔽 = ℝ then the converse is also true.
   • Note that a function f is absolutely homogeneous if and only if - f is absolutely homogeneous. However, in practice, real-valued absolutely homogeneous functions are almost always non-negative.

Definition: We say that a map L : X → Y between two vector spaces over the same scalar field 𝔽 is linear if it is additive and homogeneous.

Definition: We say that a map L : X → Y between two real or complex vector spaces (possibly over different scalar fields) is linear over ℝ or ℝ-linear if it is additive and real homogeneous.

Let f be a real or complex valued function on X. Then we say that f is/is a:

1. Sublinear: if f : X → ℝ is real-valued, subadditive, and nonnegative homogeneous.
   • The map on X defined by q(x) := max { f(x), f(-x) } is a seminorm on X called the associated seminorm.[4]
   • Every sublinear function is a convex functional that satisfies f(0) = 0, f(x) - f(y) ≤ f(x - y) and 0 ≤ max {f(x), f(-x) } for all x, y ∈ X.[3][4]
   • The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem.[3]
2. Seminorm: if f : X → ℝ is a nonnegative, real-valued, absolutely homogeneous, sublinear function.
3. Norm: if f : X → ℝ is a positive definite seminorm.
   • Every norm is a seminorm and a sublinear function.
4. Real linear functional: if f : X → ℝ is real-valued and linear over ℝ.
   • Note that this property is independent of whether the underlying scalar field of X (i.e. 𝔽) is ℝ or ℂ.
5. Linear functional: if f : X → 𝔽 is linear and takes value in 𝔽 (where recall that 𝔽 is the underlying scalar field of X).
   • If 𝔽 = ℂ then the only linear functional on X that is also a real linear functional on X is the trivial map. If 𝔽 = ℝ then linear functionals and real linear functionals are completely identical.
   • It is emphasized that 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), unless indicated otherwise.
6. Trivial: if f is identically equal to 0.

Definition: If S is a subset of X and if f : S → ℝ is a function then we say that p dominates f on S or that f is bounded above by p 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.

Examples and relationships between maps

• A sublinear function is real valued and could possibly take on negative values if it is not assumed to be positive (i.e. non-negative).
  • In contrast, a seminorm and a norm can only take on non-negative real values.
  • A non-trivial linear functional or non-trivial real linear functional necessarily takes on some negative values.
• Examples of sublinear functions: every seminorm, norm, real linear functional, and linear functional is a sublinear function.
  • Moreover, every norm and seminorm is a nonnegative (i.e. positive) ℝ-valued sublinear function.
  • Minkowski functionals of convex sets are a useful class of sublinear functions.
• Examples of seminorms: every norm is a seminorm. The absolute value of a (real or complex) linear functional is a seminorm.
• Unless indicated otherwise, it is assumed that no function takes on the value of ±∞ (this applies, in particular, to Minkowski functionals).
• Every non-trivial real linear functional or linear functional is surjective (see footnote for proof).[5]

Topological definitions

We will assume that X and Y are topological vector spaces (TVSs) over the field 𝔽, where 𝔽 is either the real numbers ℝ or complex numbers ℂ. Note that every normed and Banach space is a topological vector space (in fact, they are Hausdorff metrizable locally convex TVSs).

Notation: For any subset S of X and any scalar a, let a S := { a s : s ∈ S}.

Bounded subsets

Definition: A subset S of a TVS X is bounded or von Neumann bounded if for every neighborhood V of the origin there exists a real number r > 0 such that S ⊆ a V for all scalars a satisfying |a| ≥ r;[6].

Definition: A subset S of a normed space (X, p) (with norm p) is norm-bounded if there exists a real number r > 0 such that p (s) ≤ r for all s ∈ S.[6].

• A subset of a normed space is norm-bounded if and only if it is (von Neumann) bounded. So these notions are identical for normed spaces.

Definition: A subset S of a metrizable TVS (X, d) with metric d is d-bounded or metrically-bounded if there exists a real number R > 0 such that d(x, y) ≤ R for all x, y ∈ S.[6]

• A (von Neumann) bounded subset of a metrizable TVS (X, d) is d-metrically bounded. The converse is in general false but it is true when X is also locally convex.[6]
• In particular, a subset of a normed or Banach space is metrically bounded if and only if it is (von Neumann) bounded. So these notions are identical for normed spaces.

Bounded vs. continuous operators

Definition: A linear map between two TVSs is called bounded if it maps (von Neumann) bounded subsets of the domain to (von Neumann) bounded subsets of the codomain.

• Every continuous linear map between two topological vector spaces (TVSs) (including every continuous linear functional) is bounded. The converse is not true in general but there are conditions under which it holds.
• For a map between two normed or Banach spaces, this is equivalent to the usual definition of a "bounded linear operator". Moreover, in this case it is also equivalent to the condition that it map norm-bounded subsets to norm-bounded subsets.
• A linear functional on a normed or Banach space is continuous if and only if it is bounded.
• If X a is metrizable TVS then a linear map from X into any other TVS is continuous if and only if it is bounded map.

Hahn-Banach theorem - Linear extensions

Hahn–Banach dominated extension theorem[3](Rudin 1991, Th. 3.2) harv error: no target: CITEREFRudin1991 (help) — If p : X → ℝ is a sublinear function on a real vector space X, and f : M → ℝ is a linear functional on a linear subspace M ⊆ X that 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

F(m) = f(m)     for all m ∈ M,

|F(x)| ≤ p(x)     for all x ∈ 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,

|f(m)| ≤ p(m)     for all m ∈ 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

F(m) = f(m)     for all m ∈ M,

|F(x)| ≤ p(x)     for all x ∈ 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.

Relaxing conditions

It is possible to relax slightly the subadditivity condition on p, requiring only that (Reed and Simon, 1980) for all x, y ∈ X and all scalars a and b satisfying |a| + |b| ≤ 1,

p(ax + by) ≤ |a| p(x) + |b| p(y) .

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.[7]

Proof

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[3] (One-dimensional dominated extension theorem) — 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.

Proof —

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.

Relation to 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. The usual proof of the Hahn–Banach theorem for the case of an infinite dimensional space X uses Zorn's lemma or, equivalently, the axiom of choice. 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:[8]

(∗): 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.[9] Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.[10]

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.[11][12]

Applying the Hahn-Banach theorem

Sometimes X is assumed to have additional structure, such as a topology or a norm, but many of the statements are purely algebraic. Unless indicated otherwise, it is to be assumed that X is a vector space over the real or the complex numbers.

General template

The following is a prototypical example of the Hahn-Banach theorem.

Hahn–Banach theorem[3] — If p : X → ℝ is a sublinear function on a real vector space X and f : M → ℝ is a linear functional on a linear subspace M ⊆ X that 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.

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:

X is a vector space, p is a sublinear function on X (possibly a seminorm), M is a vector subspace of X (possibly closed), 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.

Deducing continuity

Unless indicated otherwise f and F will be linear functionals such that F is a linear extension of f and p will be a sublinear functional. We now describe some properties that are useful in understanding and applying the Hahn-Banach theorem.

Fundamental theorem for deducing continuity

Theorem[3] — If X is a real topological vector space (TVS) (e.g. a normed space), f is a (real) linear functional on X, and p is a continuous sublinear functional on X, then f ≤ p on X implies that f is continuous.

Note in particular that if f is real-valued then |f| ≤ p implies f ≤ p while if f is complex valued then |f| ≤ p implies Re f ≤ p.

• A type of converse to this result also holds: if f is a continuous linear functional then its absolute value p := |f| is a continuous seminorm (which is a type of sublinear function) such that |f| ≤ p. This statement is true in any TVS; local convexity is not required.
• Also, if p is a sublinear functional on a real vector space X then there exists a (real) linear functional f on X such that f ≤ p.[3]

This shows how we may apply a purely algebraic version of the Hahn-Banach theorem to the case where X is a topological vector space (TVS) (e.g. a normed space):

By choosing the sublinear function p to be continuous, the inequality f ≤ p (if f is real-valued), Re f ≤ p, or |f| ≤ p will allow us to conclude that the linear functional f is continuous.

Relationships between inequalities

We now give conditions for deducing "|f| ≤ p".

• If X is a real vector space, f is a (real) linear functional on X, and p is a sublinear functional on X, then f ≤ p on X if and only if f ^-1(1) ∩ { x ∈ X : p(x) < 1 } = ∅.[3]
• If f is a ℝ-valued linear functional and p is a seminorm then f ≤ p implies that |f| ≤ p.[3]
• If f is a linear functional on a real or complex vector space X and if p is a seminorm on X, then |f| ≤ p on X if and only if Re f ≤ p on X (see footnote for a proof).[13][14]
• Note that the condition |f| ≤ p is only possible if the sublinear function p is valued in the non-negative real numbers.

Deducing continuity

• Continuity of sublinear functions: Suppose X is a TVS over the real or complex numbers and p is a sublinear functional on X. Then p is continuous if and only if it is continuous at the origin, which happens if and only if p is uniformly continuous on X. If p is positive then p is continuous if and only if { x ∈ X : p(x) < 1 } is an open subset of X.
  • This characterization also applies to seminorms and real linear functionals since they are both sublinear functions.
• If f(x) = R(x) + i I(x) is a (complex-valued) linear functional on a complex vector space X then f is bounded (resp. continuous) if and only if R = Re f is bounded (resp. continuous), or equivalently, if and only if I = Im f is bounded (resp. continuous).
• Note that if the sublinear function p is continuous then the assumption "f ≤ p on M" also allows us to conclude that the linear functional f is continuous.
• Suppose f is a linear functional on a proper vector subspace M of X (i.e. M ≠ X) and that F is a linear extension of f to X. If F is continuous (resp. bounded) then so is f (since restrictions of continuous or bounded maps have the same property). Said differently, if f is not continuous (resp. bounded) then it is impossible for F (or any other extension of f to X) to be continuous (resp. bounded). For this reason, f may be assumed to be continuous in the hypotheses of some version of the Hahn-Banach theorem.

Other observations on applying the Hahn-Banach theorem

Assumption that the subspace is closed

Note that because a continuous linear functional is always uniformly continuous, it is always possible to extend a continuous linear functional f defined on a vector subspace M of a TVS X to a unique continuous linear functional on the closure of M in X, some of these Hahn-Banach theorems assume (without loss of generality) that M is a closed vector subspace of X.

Real vs. complex vector spaces and functions

• 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]

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 ℝ-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. If the linear functional f is ℝ-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.

Example applications

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.

Proposition — Suppose X is a Hausdorff locally convex TVS over the field 𝕂 and Y is a vector subspace of X that is TVS-isomorphic to 𝕂^I for some set I. Then Y is a closed and complemented vector subspace of X.

Proof —

Since 𝕂^I is a complete TVS so is Y, and since any complete subset of a Hausdorff TVS is closed, Y is a closed subset of X. Let f = ( f_i )_i∈I : Y → 𝕂^I be a TVS isomorphism, so that each f_i : Y → 𝕂 is a continuous surjective linear functional. By the Hahn–Banach theorem, we may extend each f_i to a continuous linear functional F_i : X → 𝕂 on X. Let F := ( F_i )_i∈I : X → 𝕂^I so F is a continuous linear surjection such that its restriction to Y is F |_Y = ( F_i |_Y )_i∈I = ( f_i )_i∈I = f. It follows that if we define P := f ^-1 ∘ F : X → Y then the restriction to Y of this continuous linear map P |_Y : Y → Y is the identity map 𝟙_Y on Y, for P |_Y = f ^-1 ∘ F|_Y = f ^-1 ∘ f = 𝟙_Y. So in particular, P is a continuous linear projection onto Y (i.e. P ∘ P = P). Thus Y is complemented in X and X = Y ⊕ ker P in the category of TVSs. ∎

One may use the above result to show that every closed vector subspace of ℝ^ℕ is complemented and either finite dimensional or else TVS-isomorphic to ℝ^ℕ.

Consequences

Topological vector spaces

If X is a topological vector space, not necessarily Hausdorff or locally convex, then there exists a non-zero continuous linear form if and only if X contains a nonempty, proper, convex, open set M.[15] So if the continuous dual space of X, X*, is non-trivial then by considering X with the weak topology induced by X*, X becomes a locally convex topological vector space with a non-trivial topology that is weaker than original topology on X. If in addition, X* separates points on X (which means that for each x ∈ X there is a linear functional in X* that's non-zero on x) then X 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.

Continuous extensions on locally convex spaces[3] — 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.

Theorem[3] — A U be a convex balanced neighborhood of 0 in a locally convex topological vector space X and suppose x ∈ X is not an element of U. Then there exists a continuous linear functional f on X such that

sup |f(U)| ≤ |f(x)|.

Theorem[3] — A topological vector space X has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of 0.

Sublinear functions

Theorem[3] — If p is a sublinear function 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 function 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.

Normed spaces

• 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 |f| > |f|.[3]

The dual space C[a, b]*

We have the following consequence of the Hahn–Banach theorem.

Proposition — Let −∞ < a < b < ∞. Then, F ∈ C[a, b]* if and only if there exists a (complex) measure ρ : [a, b] → ℝ of bounded variation such that

${\displaystyle F(u)=\int _{a}^{b}u(x)d\rho (x),}$

for all u ∈ C[a, b]. In addition, ||F|| = X(ρ), where X(ρ) denotes the total variation of ρ.

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 Pu = f for u, with f given in some Banach space X. If we have control on the size of u in terms of ${\displaystyle ||F||_{X}}$ 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: ${\displaystyle (f,g)=(u,P^{*}g)}$. 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 X. We can then reasonably view this functional as a weak solution to the equation.

Characterizing reflexive Banach spaces

Theorem[16] — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.

Hahn-Banach extension property

Let X be a TVS. Say that a vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X.[17] Say that X has the Hahn-Banach extension property (HBEP) if every vector subspace of X has the extension property.[17]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

Theorem[17] (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

If a vector space X has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.[17]

A vector subspace M of a TVS X has the separation property if for every element of X such that x ∉ M, there exists a continuous linear functional f on X such that f(x) ≠ 0 and f(m) = 0 for all m ∈ M.[17] Clearly, the continuous dual space of a TVS X separates points on X if and only if { 0 } has the separation property. In 1992, Kakol proved that any infinite dimensional vector space X, there exist TVS-topologies on X that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on X.[17] However, if X is a TVS then every vector subspace of X has the extension property if and only if every vector subspace of X has the separation property.[17]

Characterizations of when extensions exist

The extension principle

The following theorem characterizes when any scalar function on X (not necessarily linear) has a continuous linear extension to all of X.

Theorem (The extension principle[18]) — Let f a scalar-valued function on a subset S of a topological vector space X. Then there exists a continuous linear functional F on X extending f if and only if there exists a continuous seminorm p on X such that

| ∑ⁿ
_i=1 a_ig(s_i)| ≤ p( ∑ⁿ
_i=1 a_is_i)

for all positive integers n and all finite sequences (a_i)ⁿ
_i=1 of scalars and elements (s_i)ⁿ
_i=1 of S.

Mazur-Orlicz theorem

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 function 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;

1. 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, ${\displaystyle \sum _{i=1}^{n}s_{i}r\left(t_{i}\right)\leq p\left(\sum _{i=1}^{n}s_{i}v\left(t_{i}\right)\right)}$.

Geometric Hahn–Banach - Hahn–Banach separation theorems

See also: Hyperplane separation theorem

Hahn–Banach separation theorems are the geometrical versions of the Hahn–Banach Theorem.[19][20] They have numerous uses in convex geometry,[21] optimization theory, and economics. The separation theorem is derived from the original form of the theorem.

Definitions

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 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 ${\displaystyle A\cup B\not \subseteq H}$;[19]
• 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 = ∅);[19]
• 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.[19]

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.[19] 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]

Characterizations of when separating functionals exist

Many of the separation theorems below may be generalized to the following theorem:

Theorem[19] — Let A and B be non-empty convex subsets of a topological vector space X.

• A and B are strongly separated by a closed hyperplane if and only if there exists a convex open neighborhood U of 0 in X such that U ∩ (B - A) = ∅ (or equivalently, if (A + U) ∩ B = ∅).[3]
• If ${\displaystyle \operatorname {Int} \left(A-B\right)\neq \emptyset }$ and X is a vector space over the reals then:

${\displaystyle 0\notin \operatorname {Int} \left(A-B\right)}$ if and only if there exists some continuous ${\displaystyle f\in X^{*}}$ such that ${\displaystyle \operatorname {sup} f(A)\leq \operatorname {inf} f(B)}$ and ${\displaystyle f\neq 0}$.[19]

• If X is locally convex then:

${\displaystyle \,0\notin \operatorname {Cl} \left(A-B\right)\,}$ if and only if A and B are strongly separated by a closed hyperplane (i.e. if there exists some continuous ${\displaystyle f\in X^{*}}$ such that ${\displaystyle \operatorname {sup} f(A)<\operatorname {inf} f(B)}$). (Such an f will necessarily be non-${\displaystyle 0}$).[3][19]

Separating ...

Vector subspaces and convex open sets

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

Geometric Hahn–Banach theorem[22] — Theorem. Let K be a convex set having a nonempty interior in a real normed linear vector space X. Suppose V is a linear subspace in X containing no interior points of K. Then there is a closed hyperplane in X containing V 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 ∈ V 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:

Theorem[23] — 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 = ∅.

Subsets of a half space

Theorem[3] — Let X be a real TVS, f a non-0 continuous real-valued linear functional on X, s a real number, H = ${\displaystyle f^{-1}(s)}$, and G a subset of X having non-empty interior. If G is a subset of a half space (i.e. either the lower half space or the lower half space) then the closure of G also lies in that half space and furthermore, the interior of G lies in the corresponding strict half space.

Sets

Theorem[3] — Set 𝕂 = ℝ or C and let X be a topological vector space over 𝕂. If A, B are convex non-empty disjoint subsets of X, then:

• If A is open then A and B are separated by a closed hyperplane; explicitly, this means that there exists a continuous linear map f : X → 𝕂 and s ∈ ℝ such that Re(f(a)) < s ≤ Re(f(b)) for all a ∈ A, b ∈ B.
• If A and B are open then A and B are strictly separated by a closed hyperplane; explicitly, this means that there exists a continuous linear map f : X → 𝕂 and s ∈ ℝ such that Re(f(a)) < s < Re(f(b)) for all a ∈ A, b ∈ B.[3]
• If X is locally convex, A is compact, and B closed, then there exists a continuous linear map f : X → 𝕂 and s, t ∈ ℝ such that Re(f(a)) < t < s < Re(f(b)) for all a ∈ A, b ∈ B.

The following theorem may be used if the sets are not necessarily disjoint.

Theorem[19] — Let X be a real locally convex topological vector space and let A and B be non-empty convex subsets. If Int A ≠ ∅ and B ∩ Int A = ∅ then there exists a continuous linear functional f on X such that sup f(A) ≤ inf f(B) and f(a) < int f(B) for all a ∈ Int A (such is f is necessarily non-zero).

Theorem[3] (Separation of a subspace and an open convex set) — Let M be a vector subspace of a topological vector space X and let U be a non-empty open convex subset of X that does not intersect M. Then there exists a continuous linear functional f on X such that f(m) = 0 for all m ∈ M and Re f > 0 on U, where Re f is the real part of f (if f is real-valued then Re f = f).

Points and sets

The following is the Hahn–Banach separation theorem for a point and a set.

Theorem[19] — Let X be a real topological vector space and A ⊆ X be convex with Int A ≠ ∅. If ${\displaystyle a_{0}\in A\setminus \operatorname {Int} A}$ then a₀ is a support point of A.

Corollary[19] — Let X be a real topological vector space, A a non-empty convex open subset of X, and x₀ ∉ A. Then there exists a continuous linear functional f on X such that ${\displaystyle f\left(a-x_{0}\right)>0}$ for every a ∈ A.

Closed sets and compact sets

Theorem[3] — Let A and B be non-empty disjoint convex subsets of a locally convex topological vector space X over the real or compact numbers. If A is closed and B is compact then they are strongly separated by a closed hyperplane (i.e. there exists a continuous real-valued linear functional f on X such that sup f(A) < inf f(B)).

Corollary — The continuous dual space of a Hausdorff locally convex TVS X separates points[24] on X.

Points and disked neighborhoods of 0

Theorem[3] — Let U be a convex balanced neighborhood of 0 in a real or complex locally convex TVS X and z ∈ X be a point not in U. There exists a continuous real-valued linear functional f on X such that sup f(U) < f(z).

Generalizations

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

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.

Hahn-Banach theorem for seminorms

Hahn-Banach theorem for seminorms[25][26] (Extending seminorms) — If M is a vector subspace of X, p is a seminorm on M, and r is a seminorm on X such that p ≤ q|_M, then there exists a seminorm P on X such that P|_M = p and P ≤ q. (see footnote for proof)[27]

Hahn-Banach sandwich theorem

Hahn-Banach sandwich theorem[3] — Let S be any subset of a real vector space X, let p be a sublinear function 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.

Maximal linear extension

Theorem[3] (Andenaes, 1970) — Let M be a vector subspace of a real vector space X, p be a sublinear function 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 extending f and satisfying G ≤ p on X, then G ≥ F implies that G = F on S.

Vector valued Hahn-Banach

See also: Vector-Valued Hahn-Banach theorems

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.

See also

• Farkas' lemma
• Fichera's existence principle
• M. Riesz extension theorem
• Separating axis theorem
• Vector-Valued Hahn-Banach theorems

Notes

1. O'Connor, John J.; Robertson, Edmund F., "Hahn–Banach theorem", MacTutor History of Mathematics archive, University of St Andrews.
2. See M. Riesz extension theorem. According to Gȧrding, L. (1970). "Marcel Riesz in memoriam". Acta Math. 124 (1): I–XI. doi:10.1007/bf02394565. MR 0256837.CS1 maint: ref=harv (link), the argument was known to Riesz already in 1918.
3. Narici 2011, pp. 177-220. sfn error: no target: CITEREFNarici2011 (help)
4. Narici 2011, pp. 120-121. sfn error: no target: CITEREFNarici2011 (help)
5. Let f be a non-trivial linear functional or real linear functional, let x ∈ X be such that s := f (x) is not equal to 0, and let y = 1/s x so that f (y) = 1/s f (x) = 1. Then for any a in the codomain of f, f (a y) = a f (y) = a, which proves that f is surjective.
6. Narici 2011, pp. 156-175. sfn error: no target: CITEREFNarici2011 (help)
7. HAHNBAN file
8. Schechter, Eric. Handbook of Analysis and its Foundations. p. 620.
9. Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (PDF). Fundamenta Mathematicae. 138: 13–19.CS1 maint: ref=harv (link)
10. Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox". Fundamenta Mathematicae. 138: 21–22.
11. Brown, D. K.; Simpson, S. G. (1986). "Which set existence axioms are needed to prove the separable Hahn–Banach theorem?". Annals of Pure and Applied Logic. 31: 123–144. doi:10.1016/0168-0072(86)90066-7.CS1 maint: ref=harv (link) Source of citation.
12. Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR2517689
13. Obvious if X is a real vector space. For the non-trivial direction, assume that Re f ≤ p on X and let x ∈ X. Let r ≥ 0 and t be real numbers such that f(x) = re^it. Then |f(x)| = r = f(e^-itx) = Re (f(e^-itx)) ≤ p(e^-itx) = p(x).
14. Wilansky 2013, p. 20. sfn error: no target: CITEREFWilansky2013 (help)
15. Schaefer 1999, p. 47
16. Narici 2011, p. 212. sfn error: no target: CITEREFNarici2011 (help)
17. Narici 2011, pp. 225-273. sfn error: no target: CITEREFNarici2011 (help)
18. Edwards 1995, pp. 124-125. sfn error: no target: CITEREFEdwards1995 (help)
19. Zălinescu, C. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 5–7. ISBN 981-238-067-1. MR 1921556.
20. Gabriel Nagy, Real Analysis lecture notes
21. Harvey, R.; Lawson, H. B. (1983). "An intrinsic characterisation of Kähler manifolds". Invent. Math. 74 (2): 169–198. doi:10.1007/BF01394312.CS1 maint: ref=harv (link)
22. Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. pp. 131–134. ISBN 978-0-471-18117-0.
23. Treves, p. 184
24. To separate points on X means that for every non-zero x ∈ X, there exists some continuous linear functional f on X such that f(x) ≠ 0.
25. Wilansky 2013, pp. 18-21. sfn error: no target: CITEREFWilansky2013 (help)
26. Narici 2011, pp. 150. sfn error: no target: CITEREFNarici2011 (help)
27. Let S be the convex hull of { m ∈ M : p(x) ≤ 1 } ∪ { x ∈ X : q(x) ≤ 1 }. Note that S is an absorbing disk in X so let q be the Minkowski functional of S. Then p = P on M and P ≤ q on X.

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