Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis. 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.

HistoryEdit

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   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 ( ) had an extension of the same norm. 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). 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. 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, find it in terms of those moments. 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, again, find it if so.

Riesz and Helly solved the problem for certain classes of spaces (such as   and   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:[3]

(The vector problem) Given a collection   of bounded linear functionals on a normed space   and a collection of scalars   determine if there is an   such that   for all  

If   happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem:[3]

(The functional problem) Given a collection   of vectors in a normed space   and a collection of scalars   determine if there is a bounded linear functional   on   such that   for all  

Riesz went on to define   space ( ) in 1910 and the   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.[3]

Theorem[3] (The functional problem) — Let   be vectors in a real or complex normed space   and let   be scalars also indexed by  

There exists a continuous linear functional   on   such that   for all   if and only if there exists a   such that for any choice of scalars   where all but finitely many   are   the following holds:

 

The Hahn–Banach theorem can be deduced from the above theorem.[3] If   is reflexive then this theorem solves the vector problem.

Hahn–Banach theoremEdit

A real-valued function   defined on a subset   of   is said to be dominated (above) by a function   if   for every   Hence the reason why the following version of the Hahn-Banach theorem is called the dominated extension theorem.

Hahn–Banach dominated extension theorem (for real linear functionals)[4][5][6] — If   is a sublinear function (such as a norm or seminorm for example) defined on a real vector space   then any linear functional defined on a vector subspace of   that is dominated above by   has at least one linear extension to all of   that is also dominated above by  

Explicitly, if   is a sublinear function, which by definition means that it satisfies

 
and if   is a linear functional defined on a vector subspace   of   such that
 
then there exists a linear functional   such that
 
 
Moreover, if   is a seminorm then   necessarily holds for all  

The theorem remains true if the requirements on   are relaxed to require only that   be a convex function:[7][8]

 
A function   is convex and satisfies   if and only if   for all vectors   and all non-negative real   such that   Every sublinear function is a convex function.

If   is linear then   if and only if[4]

 
which is the (equivalent) conclusion that some authors[4] write instead of   It follows that if   is also symmetric, meaning that   holds for all   then   if and only   Every norm is a seminorm 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   on   is an example of a sublinear function that is not a seminorm.

For complex or real vector spacesEdit

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.

Hahn–Banach theorem[3][9] — Suppose   a seminorm on a vector space   over the field   which is either   or   If   is a linear functional on a vector subspace   such that

 
then there exists a linear functional   such that
 
 

The theorem remains true if the requirements on   are relaxed to require only that for all   and all scalars   and   satisfying  [8]

 
This condition holds if and only if   is a convex and balanced function satisfying   or equivalently, if and only if it is convex, satisfies   and   for all   and all unit length scalars  

A complex-valued functional   is said to be dominated by   if   for all   in the domain of   With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly:

Hahn–Banach dominated extension theorem: If   is a seminorm defined on a real or complex vector space   then every dominated linear functional defined on a vector subspace of   has a dominated linear extension to all of   In the case where   is a real vector space and   is merely a convex or sublinear function, this conclusion will remain true if both instances of "dominated" (meaning  ) are weakened to instead mean "dominated above" (meaning  ).[7][8]

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   on a complex vector space is completely determined by its real part   through the formula[6][proof 1]

 
and moreover, if   is a norm on   then their operator norms are equal:  [10] In particular, a linear functional on   extends another one defined on   if and only if their real parts are equal on   (in other words, a linear functional   extends   if and only if   extends  ). The real part of a linear functional on   is always a real-linear functional (meaning that it is linear when   is considered as a real vector space) and if   is a real-linear functional on a complex vector space then   defines the unique linear functional on   whose real part is  

If   is a linear functional on a (complex or real) vector space   and if   is a seminorm then[6][proof 2]

 
Stated in simpler language, a linear functional is dominated by a seminorm   if and only if its real part is dominated above by  
Proof of Hahn–Banach for complex vector spaces by reduction to real vector spaces[3]

Suppose   is a seminorm on a complex vector space   and let   be a linear functional defined on a vector subspace   of   that satisfies   on   Consider   as a real vector space and apply the Hahn–Banach theorem for real vector spaces to the real-linear functional   to obtain a real-linear extension   that is also dominated above by   which means that it satisfies   on   and   on   The map   defined by   is a linear functional on   that extends   (because their real parts agree on  ) and satisfies   on   (because   and   is a seminorm).  

The proof above shows that when   is a seminorm then there is a one-to-one correspondence between dominated linear extensions of   and dominated real-linear extensions of   the proof even gives a formula for explicitly constructing a linear extension of   from any given real-linear extension of its real part.

Continuity

A linear functional   on a topological vector space is continuous if and only if this is true of its real part   if the domain is a normed space then   (where one side is infinite if and only if the other side is infinite).[10] Assume   is a topological vector space and   is sublinear function. If   is a continuous sublinear function that dominates a linear functional   then   is necessarily continuous.[6] Moreover, a linear functional   is continuous if and only if its absolute value   (which is a seminorm that dominates  ) is continuous.[6] In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function.

ProofEdit

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   to a larger vector space in which   has codimension  [3]

Lemma[6] (One–dimensional dominated extension theorem) — Let   be a sublinear function on a real vector space   let   a linear functional on a proper vector subspace   such that   on   (meaning   for all  ), and let   be a vector not in   (so  ). There exists a linear extension   of   such that   on  

Proof[6]

Given any real number   the map   defined by   is always a linear extension of   to  [note 1] but it might not satisfy   It will be shown that   can always be chosen so as to guarantee that   which will complete the proof.

If   then

 
which implies
 
So define
 
where   are real numbers. To guarantee   it suffices that   (in fact, this is also necessary[note 2]) because then   satisfies "the decisive inequality"[6]
 

To see that   follows,[note 3] assume   and substitute   in for both   and   to obtain

 
If   (respectively, if  ) then the right (respectively, the left) hand side equals   so that multiplying by   gives    

This lemma remains true if   is merely a convex function instead of a sublinear function.[7][8]

Proof

Assume that   is convex, which means that   for all   and   Let     and   be as in the lemma's statement. Given any   and any positive real   the positive real numbers   and   sum to   so that the convexity of   on   guarantees

 
and hence
 
thus proving that   which after multiplying both sides by   becomes
 
This implies that the values defined by
 
are real numbers that satisfy   As in the above proof of the one–dimensional dominated extension theorem above, for any real   define   by   It can be verified that if   then   where   follows from   when   (respectively, follows from   when  ).  

The lemma above is the key step in deducing the dominated extension theorem from Zorn's lemma.

Proof of dominated extension theorem using Zorn's lemma

The set of all possible dominated linear extensions of   are partially ordered by extension of each other, so there is a maximal extension   By the codimension-1 result, if   is not defined on all of   then it can be further extended. Thus   must be defined everywhere, as claimed.  

When   has countable codimension, then using induction and the lemma completes the proof of the Hahn–Banach theorem. The standard proof of the general case uses Zorn's lemma although the strictly weaker ultrafilter lemma[11] (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 spaces[12] (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.[13]

In locally convex spacesEdit

In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question. However, in locally convex spaces, any continuous functional is already bounded by the norm, which is sublinear. One thus has

Continuous extensions on locally convex spaces[3] — Let X be locally convex topological vector space over   (either   or  ),   a vector subspace of X, and   a continuous linear functional on   Then   has a continuous linear extension to all of X. If the topology on X arises from a norm, then the norm of   is preserved by this extension.

In category-theoretic terms, the field   is an injective object in the category of locally convex vector spaces.

Geometric Hahn–Banach (the Hahn–Banach separation theorems)Edit

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,[14] optimization theory, and economics. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems.[15][16]

Theorem[15] — Let   and   be non-empty convex subsets of a real locally convex topological vector space   If   and   then there exists a continuous linear functional   on   such that   and   for all   (such an   is necessarily non-zero).

When the convex sets have additional properties, such as being open or compact for example, then the conclusion can be substantially strengthened:

Theorem[3][17] — Let   and   be convex non-empty disjoint subsets of a real topological vector space  

  • If   is open then   and   are separated by a closed hyperplane. Explicitly, this means that there exists a continuous linear map   and   such that   for all   If both   and   are open then the right-hand side may be taken strict as well.
  • If   is locally convex,   is compact, and   closed, then   and   are strictly separated: there exists a continuous linear map   and   such that   for all  

If   is complex (rather than real) then the same claims hold, but for the real part of  

Then following important corollary is known as the Geometric Hahn–Banach theorem or Mazur's theorem. It follows from the first bullet above and the convexity of  

Theorem (Mazur)[18] — Let   be a vector subspace of the topological vector space   and suppose   is a non-empty convex open subset of   with   Then there is a closed hyperplane (codimension-1 vector subspace)   that contains   but remains disjoint from  

Mazur's theorem clarifies that vector subspaces (even those that are not closed) can be characterized by linear functionals.

Corollary[19] (Separation of a subspace and an open convex set) — Let   be a vector subspace of a locally convex topological vector space   and   be a non-empty open convex subset disjoint from   Then there exists a continuous linear functional   on   such that   for all   and   on  

Supporting hyperplanesEdit

Since points are trivially convex, geometric Hahn-Banach implies that functionals can detect the boundary of a set. In particular, let   be a real topological vector space and   be convex with   If   then there is a functional that is vanishing at   but supported on the interior of  [15]

Call a normed space   smooth if at each point   in its unit ball there exists a unique closed hyperplane to the unit ball at   Köthe showed in 1983 that a normed space is smooth at a point   if and only if the norm is Gateaux differentiable at that point.[3]

Balanced or disked neighborhoodsEdit

Let   be a convex balanced neighborhood of the origin in a locally convex topological vector space   and suppose   is not an element of   Then there exists a continuous linear functional   on   such that[3] 

ApplicationsEdit

The Hahn–Banach theorem is the first sign of an important philosophy in functional analysis: to understand a space, one should understand its continuous functionals.

For example, linear subspaces are characterized by functionals: if X is a normed vector space with linear subspace M (not necessarily closed) and if   is an element of X not in the closure of M, then there exists a continuous linear map   with   for all     and   (To see this, note that   is a sublinear function.) Moreover, if   is an element of X, then there exists a continuous linear map   such that   and   This implies that the natural injection   from a normed space X into its double dual   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 X is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set M. Then geometric Hahn-Banach implies that there is a hyperplane separating M from any other point. In particular, there must exist a nonzero functional on X — that is, the continuous dual space   is non-trivial.[3][20] Considering X with the weak topology induced by   then X becomes locally convex; by the second bullet of geometric Hahn-Banach, the weak topology on this new space separates points. Thus 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.

Partial differential equationsEdit

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   for   with   given in some Banach space X. If we have control on the size of   in terms of   and we can think of   as a bounded linear functional on some suitable space of test functions   then we can view   as a linear functional by adjunction:   At first, this functional is only defined on the image of   but using the Hahn–Banach theorem, we can try to extend it to the entire codomain X. The resulting functional is often defined to be a weak solution to the equation.

Characterizing reflexive Banach spacesEdit

Theorem[21] — 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.

Example from Fredholm theoryEdit

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   is a Hausdorff locally convex TVS over the field   and   is a vector subspace of   that is TVS–isomorphic to   for some set   Then   is a closed and complemented vector subspace of  

Proof

Since   is a complete TVS so is   and since any complete subset of a Hausdorff TVS is closed,   is a closed subset of   Let   be a TVS isomorphism, so that each   is a continuous surjective linear functional. By the Hahn–Banach theorem, we may extend each   to a continuous linear functional   on   Let   so   is a continuous linear surjection such that its restriction to   is   Let   which is a continuous linear map whose restriction to   is   where   denotes the identity map on   This shows that   is a continuous linear projection onto   (that is,  ). Thus   is complemented in   and   in the category of TVSs.  

The above result may be used to show that every closed vector subspace of   is complemented because any such space is either finite dimensional or else TVS–isomorphic to  

GeneralizationsEdit

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:

  is a sublinear function (possibly a seminorm) on a vector space     is a vector subspace of   (possibly closed), and   is a linear functional on   satisfying   on   (and possibly some other conditions). One then concludes that there exists a linear extension   of   to   such that   on   (possibly with additional properties).

Theorem[3] — If   is an absorbing disk in a real or complex vector space   and if   be a linear functional defined on a vector subspace   of   such that   on   then there exists a linear functional   on   extending   such that   on  

For seminormsEdit

Hahn–Banach theorem for seminorms[22][23] — If   is a seminorm defined on a vector subspace   of   and if   is a seminorm on   such that   then there exists a seminorm   on   such that   on   and   on  

Proof of the Hahn–Banach theorem for seminorms

Let   be the convex hull of   Because   is an absorbing disk in   its Minkowski functional   is a seminorm. Then   on   and   on  

Geometric separationEdit

Hahn–Banach sandwich theorem[3] — Let   be a sublinear function on a real vector space   let   be any subset of   and let   be any map. If there exist positive real numbers   and   such that

 
then there exists a linear functional   on   such that   on   and   on  

Maximal dominated linear extensionEdit

Theorem[3] (Andenaes, 1970) — Let   be a sublinear function on a real vector space   let   be a linear functional on a vector subspace   of   such that   on   and let   be any subset of   Then there exists a linear functional   on   that extends   satisfies   on   and is (pointwise) maximal on   in the following sense: if   is a linear functional on   that extends   and satisfies   on   then   on   implies   on  

If   is a singleton set (where   is some vector) and if   is such a maximal dominated linear extension of   then  [3]

Vector valued Hahn–BanachEdit

Vector–valued Hahn–Banach theorem[3] — If   and   are vector spaces over the same field and if   be a linear map defined on a vector subspace   of   then there exists a linear map   that extends  

For nonlinear functionsEdit

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

Mazur–Orlicz theorem[3] — Let   be a sublinear function on a real or complex vector space   let   be any set, and let   and   be any maps. The following statements are equivalent:

  1. there exists a real-valued linear functional   on   such that   on   and   on  ;
  2. for any finite sequence   of   non-negative real numbers, and any sequence   of elements of  
     

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

Theorem (The extension principle[24]) — Let   a scalar-valued function on a subset   of a topological vector space   Then there exists a continuous linear functional   on   extending   if and only if there exists a continuous seminorm   on   such that

 
for all positive integers   and all finite sequences   of scalars and elements   of  

ConverseEdit

Let X be a topological vector space. 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, and we say that X has the Hahn–Banach extension property (HBEP) if every vector subspace of X has the extension property.[25]

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.[25] On the other hand, a vector space X 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.[25]

A vector subspace M of a TVS X has the separation property if for every element of X such that   there exists a continuous linear functional   on X such that   and   for all   Clearly, the continuous dual space of a TVS X separates points on X if and only if   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. 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.[25]

Relation to axiom of choice and other theoremsEdit

The proof of the Hahn–Banach theorem for real vector spaces (HB) commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory (ZF) is equivalent to the axiom of choice (AC). It was discovered by Łoś and Ryll-Nardzewski[12] and independently by Luxemburg[11] 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.[26]

The ultrafilter lemma is equivalent (under ZF) to the Banach–Alaoglu theorem,[27] which is another foundational theorem in functional analysis. Although the Banach–Alaoglu theorem implies HB,[28] 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.[29] The Hahn–Banach theorem is also equivalent to the following statement:[30]

(∗): On every Boolean algebra B there exists a "probability charge", that is: a non-constant finitely additive map from   into  

(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.[31] Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox.[32]

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.[33][34]

See alsoEdit

NotesEdit

  1. ^ This definition means, for instance, that   and if   then   In fact, if   is any linear extension of   to   then   for   In other words, every linear extension of   to   is of the form   for some (unique)  
  2. ^ Explicitly, for any real number     on   if and only if   Combined with the fact that   it follows that the dominated linear extension of   to   is unique if and only if   in which case this scalar will be the extension's values at   Since every linear extension of   to   is of the form   for some   the bounds   thus also limit the range of possible values (at  ) that can be taken by any of  's dominated linear extensions. Specifically, if   is any linear extension of   satisfying   then for every    
  3. ^ Geometric illustration: The geometric idea of the above proof can be fully presented in the case of   First, define the simple-minded extension   It doesn't work, since maybe  . But it is a step in the right direction.   is still convex, and   Further,   is identically zero on the x-axis. Thus we have reduced to the case of   on the x-axis. If   on   then we are done. Otherwise, pick some   such that   The idea now is to perform a simultaneous bounding of   on   and   such that   on   and   on   then defining   would give the desired extension. Since   are on opposite sides of   and   at some point on   by convexity of   we must have   on all points on   Thus   is finite. Geometrically, this works because   is a convex set that is disjoint from   and thus must lie entirely on one side of   Define   This satisfies   on   It remains to check the other side. For all   convexity implies that for all   thus   Since during the proof, we only used convexity of  , we see that the lemma remains true for merely convex  

Proofs

  1. ^ If   has real part   then   which proves that   Substituting   in for   and using   gives    
  2. ^ Let   be any homogeneous scalar-valued map on   (such as a linear functional) and let   be any map that satisfies   for all   and unit length scalars   (such as a seminorm). If   then   For the converse, assume   and fix   Let   and pick any   such that   it remains to show   Homogeneity of   implies   is real so that   By assumption,   and   so that   as desired.  

ReferencesEdit

  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., the argument was known to Riesz already in 1918.
  3. ^ a b c d e f g h i j k l m n o p q r s t Narici & Beckenstein 2011, pp. 177–220.
  4. ^ a b c Rudin 1991, pp. 56–62.
  5. ^ Rudin 1991, Th. 3.2
  6. ^ a b c d e f g h Narici & Beckenstein 2011, pp. 177–183.
  7. ^ a b c Schechter 1996, pp. 318–319.
  8. ^ a b c d Reed & Simon 1980.
  9. ^ Rudin 1991, Th. 3.2
  10. ^ a b Narici & Beckenstein 2011, pp. 126–128.
  11. ^ a b Luxemburg 1962.
  12. ^ a b Łoś & Ryll-Nardzewski 1951, pp. 233–237.
  13. ^ HAHNBAN file
  14. ^ Harvey, R.; Lawson, H. B. (1983). "An intrinsic characterisation of Kähler manifolds". Invent. Math. 74 (2): 169–198. Bibcode:1983InMat..74..169H. doi:10.1007/BF01394312. S2CID 124399104.
  15. ^ a b c 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.
  16. ^ Gabriel Nagy, Real Analysis lecture notes
  17. ^ Brezis, Haim (2011). Functional Analysis, Sobolev Spaces, and Partial Differential Equations. New York: Springer. pp. 6–7.
  18. ^ Trèves 2006, p. 184.
  19. ^ Narici & Beckenstein 2011, pp. 195.
  20. ^ Schaefer & Wolff 1999, p. 47.
  21. ^ Narici & Beckenstein 2011, p. 212.
  22. ^ Wilansky 2013, pp. 18–21.
  23. ^ Narici & Beckenstein 2011, pp. 150.
  24. ^ Edwards 1995, pp. 124–125.
  25. ^ a b c d Narici & Beckenstein 2011, pp. 225–273.
  26. ^ Pincus 1974, pp. 203–205.
  27. ^ Schechter 1996, pp. 766–767.
  28. ^ Muger, Michael (2020). Topology for the Working Mathematician.
  29. ^ Bell, J.; Fremlin, David (1972). "A Geometric Form of the Axiom of Choice" (PDF). Fundamenta Mathematicae. 77 (2): 167–170. doi:10.4064/fm-77-2-167-170. Retrieved 26 Dec 2021.
  30. ^ Schechter, Eric. Handbook of Analysis and its Foundations. p. 620.
  31. ^ Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (PDF). Fundamenta Mathematicae. 138: 13–19. doi:10.4064/fm-138-1-13-19.
  32. ^ Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox". Fundamenta Mathematicae. 138: 21–22. doi:10.4064/fm-138-1-21-22.
  33. ^ 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. Source of citation.
  34. ^ Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, ISBN 978-0-521-88439-6, MR2517689

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