Locally convex topological vector space

Summary

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

HistoryEdit

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914,[1] although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces.[2][3] Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).[4][5]

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces[6] (in which case the unit ball of the dual is metrizable).

DefinitionEdit

Suppose   is a vector space over   a subfield of the complex numbers (normally   itself or  ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Definition via convex setsEdit

A subset   in   is called

  1. Convex if for all   and     In other words,   contains all line segments between points in  
  2. Circled if for all   and scalars   if   then   If   this means that   is equal to its reflection through the origin. For   it means for any     contains the circle through   centred on the origin, in the one-dimensional complex subspace generated by  
  3. Balanced if for all   and scalars   if   then   If   this means that if   then   contains the line segment between   and   For   it means for any in     contains the disk with   on its boundary, centred on the origin, in the one-dimensional complex subspace generated by   Equivalently, a balanced set is a circled cone.
  4. A cone (when the underlying field is ordered) if for all   and    
  5. Absorbent or absorbing if for every   there exists   such that   for all   satisfying   The set   can be scaled out by any "large" value to absorb every point in the space.
    • In any TVS, every neighborhood of the origin is absorbent.[7]
  6. Absolutely convex or a disk if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to  ; such a set is absorbent if it spans all of  

A topological vector space (TVS) is called locally convex if the origin has a neighborhood basis (that is, a local base) consisting of convex sets.[7]

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets.[7] Every TVS has a neighborhood basis at the origin consisting of balanced sets but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced and convex. It is possible for a TVS to have some neighborhoods of the origin that are convex and yet not be locally convex.

Because translation is (by definition of "topological vector space") continuous, all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

Definition via seminormsEdit

A seminorm on   is a map   such that

  1.   is nonnegative or positive semidefinite:  ;
  2.   is positive homogeneous or positive scalable:   for every scalar   So, in particular,  ;
  3.   is subadditive. It satisfies the triangle inequality:  

If   satisfies positive definiteness, which states that if   then   then   is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

If   is a vector space and   is a family of seminorms on   then a subset   of   is called a base of seminorms for   if for all   there exists a   and a real   such that  [8]

Definition (second version): A locally convex space is defined to be a vector space   along with a family   of seminorms on  

Seminorm topologyEdit

Suppose that   is a vector space over   where   is either the real or complex numbers, and let   (resp.  ) denote the open (resp. closed) ball of radius   in   A family of seminorms   on the vector space   induces a canonical vector space topology on  , called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on   for which all maps in   are continuous.

That the vector space operations are continuous in this topology follows from properties 2 and 3 above. It can easily be seen that the resulting topological vector space is "locally convex" in the sense of the first definition given above because each   is absolutely convex and absorbent (and because the latter properties are preserved by translations).

It is possible for a locally convex topology on a space   to be induced by a family of norms but for   to not be normable (that is, to have its topology be induced by a single norm).

Basis and subbasesEdit

Suppose that   is a family of seminorms on   that induces a locally convex topology   on  . A subbasis at the origin is given by all sets of the form   as   ranges over   and   ranges over the positive real numbers. A base at the origin is given by the collection of all possible finite intersections of such subbasis sets.

Recall that the topology of a TVS is translation invariant, meaning that if   is any subset of   containing the origin then for any     is a neighborhood of the origin if and only if   is a neighborhood of  ; thus it suffices to define the topology at the origin. A base of neighborhoods of   for this topology is obtained in the following way: for every finite subset   of   and every   let

 
Bases of seminorms and saturated familiesEdit

If   is a locally convex space and if   is a collection of continuous seminorms on  , then   is called a base of continuous seminorms if it is a base of seminorms for the collection of all continuous seminorms on  .[8] Explicitly, this means that for all continuous seminorms   on  , there exists a   and a real   such that  [8]

If   is a base of continuous seminorms for a locally convex TVS   then the family of all sets of the form   as   varies over   and   varies over the positive real numbers, is a base of neighborhoods of the origin in   (not just a subbasis, so there is no need to take finite intersections of such sets).[8]

A family   of seminorms on a vector space   is called saturated if for any   and   in   the seminorm defined by   belongs to  

If   is a saturated family of continuous seminorms that induces the topology on   then the collection of all sets of the form   as   ranges over   and   ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;[8] This forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to take finite intersections of such sets.[8]

Basis of normsEdit

The following theorem implies that if   is a locally convex space then the topology of   can be a defined by a family of continuous norms on   (a norm is a seminorm   where   implies  ) if and only if there exists at least one continuous norm on  .[9] This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family   of seminorms (each of which is necessarily continuous) then the family   of (also continuous) norms obtained by adding some given continuous norm   to each element, will necessarily be a family of norms that defines this same locally convex topology. If there exists a continuous norm on a topological vector space   then   is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

Theorem[10] — Let   be a Fréchet space over the field   Then the following are equivalent:

  1.   does not admit a continuous norm (that is, any continuous seminorm on   can not be a norm).
  2.   contains a vector subspace that is TVS-isomorphic to  
  3.   contains a complemented vector subspace that is TVS-isomorphic to  
NetsEdit

Suppose that the topology of a locally convex space   is induced by a family   of continuous seminorms on  . If   and if   is a net in  , then   in   if and only if for all    [11] Moreover, if   is Cauchy in  , then so is   for every  [11]

Equivalence of definitionsEdit

Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their  -balls is the triangle inequality.

For an absorbing set   such that if   then   whenever   define the Minkowski functional of   to be

 

From this definition it follows that   is a seminorm if   is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

 
form a base of convex absorbent balanced sets.

Ways of defining a locally convex topologyEdit

Theorem[7] — Suppose that   is a (real or complex) vector space and let   be a filter base of subsets of   such that:

  1. Every   is convex, balanced, and absorbing;
  2. For every   there exists some real   satisfying   such that  

Then   is a neighborhood base at 0 for a locally convex TVS topology on  .

Theorem[7] — Suppose that   is a (real or complex) vector space and let   be a non-empty collection of convex, balanced, and absorbing subsets of  . Then the set of all of all positive scalar multiples of finite intersections of sets in   forms a neighborhood base at the origin for a locally convex TVS topology on  .

Further definitionsEdit

  • A family of seminorms   is called total or separated or is said to separate points if whenever   holds for every   then   is necessarily   A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
  • A pseudometric is a generalization of a metric which does not satisfy the condition that   only when   A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
     
    (where the   can be replaced by any positive summable sequence  ). This pseudometric is translation-invariant, but not homogeneous, meaning   and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
  • As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
  • A Cauchy net in a locally convex space is a net   such that for every   and every seminorm   there exists some index   such that for all indices     In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
  • A family of seminorms becomes a preordered set under the relation   if and only if there exists an   such that for all     One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every   and   there is a   such that   Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family   let   be the set of finite subsets of   and then for every   define
     
    One may check that   is an equivalent directed family.
  • If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a bounded neighborhood.

Sufficient conditionsEdit

Hahn–Banach extension propertyEdit

Let   be a TVS. Say that a vector subspace   of   has the extension property if any continuous linear functional on   can be extended to a continuous linear functional on  .[12] Say that   has the Hahn-Banach extension property (HBEP) if every vector subspace of   has the extension property.[12]

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

Theorem[12] (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

If a vector space   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.[12]

PropertiesEdit

Throughout,   is a family of continuous seminorms that generate the topology of  .

Topological propertiesEdit

  • Suppose that   is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of   are exactly those that are of the form   for some   and some positive continuous sublinear functional   on  .[13]
  • If   and   then   if and only if for every   and every finite collection   there exists some   such that  [14]
  • The closure of   in   is equal to  [15]
  • Every Hausdorff locally convex TVS is homeomorphic to a subspace of a product of Banach spaces.[16]

Topological properties of convex subsetsEdit

  • The interior and closure of a convex subset of a TVS is again convex.[17]
  • The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.[17]
  • If   is a convex set with non-empty interior, then the closure of   is equal to the closure of the interior of  ; furthermore, the interior of   is equal to the interior of the closure of  [17][18]
    • So if a convex set   has non-empty interior then   is a closed (resp. open) set if and only if it is a regular closed (resp. regular open) set.
  • If   is a convex subset of a TVS   (not necessarily Hausdorff),   belongs to the interior of  , and   belongs to the closure of  , then the open line segment joint   and   (that is,  ) belongs to the interior of  .[18][19]
  • If   is a locally convex space (not necessarily Hausdorff),   is a closed vector subspace of  , V is a convex neighborhood of the origin in  , and if   is a vector not in   then there exists a convex neighborhood U of 0 in   such that   and  [17]
  • The closure of a convex subset of a Hausdorff locally convex TVS   is the same for all locally convex Hausdorff TVS topologies on   that are compatible with duality between   and its continuous dual space.[20]
  • In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.[7]
  • In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.[7]
    • More generally, if   is a compact subset of a locally convex space, then the convex hull   (resp. the disked hull  ) is compact if and only if it is complete.[7]
  • In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.[21]
  • In a Fréchet space, the closed convex hull of a compact set is compact.[22]
  • In a locally convex space, any linear combination of totally bounded sets is totally bounded.[21]

Properties of convex hullsEdit

For any subset   of a TVS  , the convex hull (resp. closed convex hull, balanced hull, resp. convex balanced hull) of  , denoted by   (respectively,      ), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of   containing  

  • The convex hull of compact subset of a Hilbert space is not necessarily closed and so also not necessarily compact. For example, let   be the separable Hilbert space   of square-summable sequences with the usual norm   and let   be the standard orthonormal basis (that is   at the  -coordinate). The closed set   is compact but its convex hull   is not a closed set because   belongs to the closure of   in   but   (since every sequence   is a finite convex combination of elements of   and so is necessarily   in all but finitely many coordinates, which is not true of  ).[23] However, like in all complete Hausdorff locally convex spaces, the closed convex hull   of this compact subset is compact. The vector subspace   is a pre-Hilbert space when endowed with the substructure that the Hilbert space   induces on it but   is not complete and   (since  ). The closed convex hull of   in   (here, "closed" means with respect to   and not to   as before) is equal to   which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might fail to be compact (although it will be precompact/totally bounded).
  • In a Hausdorff locally convex space   the closed convex hull   of compact subset   is not necessarily compact although it is a precompact (also called "totally bounded") subset, which means that its closure, when taken in a completion   of   will be compact (here   so that   if and only if   is complete); that is to say,   will be compact. So for example, the closed convex hull   of a compact subset of   of a pre-Hilbert space   is always a precompact subset of   and so the closure of   in any Hilbert space   containing   (such as the Hausdorff completion of   for instance) will be compact (this is the case in the previous example above).
  • In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
  • In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact.[24] Consequently, in a complete Hausdorff locally convex space, the closed convex hull of a compact subset is again compact.[25]
  • In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).[7]
    • This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is closed (in addition to being compact and convex); in particular, the convex hull of such a union is equal to the closed convex hull of that union.
    • In general, the closed convex hull of a compact set is not necessarily compact.
    • In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but not closed.
  • The bipolar theorem states that the bipolar (that is, the polar of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.[26]
  • The balanced hull of a convex set is not necessarily convex.
  • If   and   are convex subsets of a topological vector space   and if   then there exist     and a real number   satisfying   such that  [17]
  • If   is a vector subspace of a TVS  ,   a convex subset of  , and   a convex subset of   such that   then  [17]
  • Recall that the smallest balanced subset of   containing a set   is called the balanced hull of   and is denoted by   For any subset   of  , the convex balanced hull of  , denoted by   is the smallest subset of   containing   that is convex and balanced.[27] The convex balanced hull of   is equal to the convex hull of the balanced hull of   (i.e.  ), but the convex balanced hull of   is not necessarily equal to the balanced hull of the convex hull of   (that is,   is not necessarily equal to  ).[27]
  • If   are subsets of a TVS   and if   is a scalar then     and   Moreover, if   is compact then   [28]
  • If   are subsets of a TVS   whose closed convex hulls are compact, then  [28]
  • If   is a convex set in a complex vector space   and there exists some   such that   then   for all real   such that   In particular,   for all scalars   such that  

Examples and nonexamplesEdit

Finest and coarsest locally convex topologyEdit

Coarsest vector topologyEdit

Any vector space   endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only   The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on   if and only   This follows from the fact that every topological vector space is a connected space.

Finest locally convex topologyEdit

If   is a real or complex vector space and if   is the set of all seminorms on   then the locally convex TVS topology, denoted by   that   induces on   is called the finest locally convex topology on  [29] This topology may also be described as the TVS-topology on   having as a neighborhood base at the origin the set of all absorbing disks in  [29] Any locally convex TVS-topology on   is necessarily a subset of     is Hausdorff.[15] Every linear map from   into another locally convex TVS is necessarily continuous.[15] In particular, every linear functional on   is continuous and every vector subspace of   is closed in  ;[15] therefore, if   is infinite dimensional then   is not pseudometrizable (and thus not metrizable).[29] Moreover,   is the only Hausdorff locally convex topology on   with the property that any linear map from it into any Hausdorff locally convex space is continuous.[30] The space   is a bornological space.[31]

Examples of locally convex spacesEdit

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the   spaces with   are locally convex.

More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.

The space   of real valued sequences with the family of seminorms given by

 
is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the limit topology of the spaces   embedded in   in the natural way, by completing finite sequences with infinitely many  

Given any vector space   and a collection   of linear functionals on it,   can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in   continuous. This is known as the weak topology or the initial topology determined by   The collection   may be the algebraic dual of   or any other collection. The family of seminorms in this case is given by   for all   in  

Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions   such that   where   and   are multiindices. The family of seminorms defined by   is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.

An important function space in functional analysis is the space   of smooth functions with compact support in   A more detailed construction is needed for the topology of this space because the space   is not complete in the uniform norm. The topology on   is defined as follows: for any fixed compact set   the space   of functions   with   is a Fréchet space with countable family of seminorms   (these are actually norms, and the completion of the space   with the   norm is a Banach space  ). Given any collection   of compact sets, directed by inclusion and such that their union equal   the   form a direct system, and   is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely,   is the union of all the   with the strongest locally convex topology which makes each inclusion map   continuous. This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of   is the space of distributions on  

More abstractly, given a topological space   the space   of continuous (not necessarily bounded) functions on   can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms   (as   varies over the directed set of all compact subsets of  ). When   is locally compact (for example, an open set in  ) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of   that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.

Examples of spaces lacking local convexityEdit

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

  • The spaces   for   are equipped with the F-norm
     
    They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces   with an atomless, finite measure   and   are not locally convex.
  • The space of measurable functions on the unit interval   (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability):
     
    This space is often denoted  

Both examples have the property that any continuous linear map to the real numbers is   In particular, their dual space is trivial, that is, it contains only the zero functional.

  • The sequence space     is not locally convex.

Continuous mappingsEdit

Theorem[32] — Let   be a linear operator between TVSs where   is locally convex (note that   need not be locally convex). Then   is continuous if and only if for every continuous seminorm   on  , there exists a continuous seminorm   on   such that  

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces   and   with families of seminorms   and   respectively, a linear map   is continuous if and only if for every   there exist   and   such that for all  

 

In other words, each seminorm of the range of   is bounded above by some finite sum of seminorms in the domain. If the family   is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

 

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

Linear functionalsEdit

Theorem[32] — If   is a TVS (not necessarily locally convex) and if   is a linear functional on  , then   is continuous if and only if there exists a continuous seminorm   on   such that  

If   is a real or complex vector space,   is a linear functional on  , and   is a seminorm on  , then   if and only if  [33] If   is a non-0 linear functional on a real vector space   and if   is a seminorm on  , then   if and only if  [15]

Multilinear mapsEdit

Let   be an integer,   be TVSs (not necessarily locally convex), let   be a locally convex TVS whose topology is determined by a family   of continuous seminorms, and let   be a multilinear operator that is linear in each of its   coordinates. The following are equivalent:

  1.   is continuous.
  2. For every   there exist continuous seminorms   on   respectively, such that   for all  [15]
  3. For every   there exists some neighborhood of the origin in   on which   is bounded.[15]

See alsoEdit

NotesEdit

  1. ^ Hausdorff, F. Grundzüge der Mengenlehre (1914)
  2. ^ von Neumann, J. Collected works. Vol II. pp. 94–104
  3. ^ Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1.
  4. ^ von Neumann, J. Collected works. Vol II. pp. 508–527
  5. ^ Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.
  6. ^ Banach, S. Theory of linear operations p. 75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932)
  7. ^ a b c d e f g h i Narici & Beckenstein 2011, pp. 67–113.
  8. ^ a b c d e f Narici & Beckenstein 2011, p. 122.
  9. ^ Jarchow 1981, p. 130.
  10. ^ Jarchow 1981, pp. 129–130.
  11. ^ a b Narici & Beckenstein 2011, p. 126.
  12. ^ a b c d Narici & Beckenstein 2011, pp. 225–273.
  13. ^ Narici & Beckenstein 2011, pp. 177–220.
  14. ^ Narici & Beckenstein 2011, p. 149.
  15. ^ a b c d e f g Narici & Beckenstein 2011, pp. 149–153.
  16. ^ Narici & Beckenstein 2011, pp. 115–154.
  17. ^ a b c d e f Trèves 2006, p. 126.
  18. ^ a b Schaefer & Wolff 1999, p. 38.
  19. ^ Conway 1990, p. 102.
  20. ^ Trèves 2006, p. 370.
  21. ^ a b Narici & Beckenstein 2011, pp. 155–176.
  22. ^ Rudin 1991, p. 7.
  23. ^ Aliprantis & Border 2006, p. 185.
  24. ^ Trèves 2006, p. 67.
  25. ^ Trèves 2006, p. 145.
  26. ^ Trèves 2006, p. 362.
  27. ^ a b Trèves 2006, p. 68.
  28. ^ a b Dunford 1988, p. 415.
  29. ^ a b c Narici & Beckenstein 2011, pp. 125–126.
  30. ^ Narici & Beckenstein 2011, p. 476.
  31. ^ Narici & Beckenstein 2011, p. 446.
  32. ^ a b Narici & Beckenstein 2011, pp. 126–128.
  33. ^ Narici & Beckenstein 2011, pp. 126-–128.

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