In the field of functional analysis, a subfield of mathematics, a dual system, dual pair, or a duality over a field ( is either the real or the complex numbers) is a triple consisting of two vector spaces over and a bilinear map such that for all non-zero the map is not identically and for all non-zero , the map is not identically 0. The study of dual systems is called duality theory.
According to Helmut H. Schaefer, "the study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the deepest and most beautiful results of the subject."[1]
Definition, notation, and conventions
A pairing or a pair over a field is a triple , which may also be denoted by , consisting of two vector spaces and over (which this article assumes is either the real numbers or the complex numbers) and is a bilinear map, which is called the bilinear map associated with the pairing[2] or simply the pairing's map/bilinear form.
For all , let denote the linear functional on defined by and let . Similarly, for all , let be defined by and let .
It is common practice to write instead of , in which case the pair is often denoted by rather than . However, this article will reserve use of for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.
A pairing is called a dual system, a dual pair,[3] or a duality over if the bilinear form is non-degenerate, which means that it satisfies the following two separation axioms:
separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero , the map is not identically (i.e. there exists a such that ;
separates/distinguishes points of : if is such that then ; or equivalently, for all non-zero , the map is not identically (i.e. there exists an such that .
In this case say that is non-degenerate, say that places and in duality (or in separated duality), and is called the duality pairing of the .[2][3]
A subset of is called total if for all , for all implies . A total subset of is defined analogously (see footnote).[note 1]
The elements and are orthogonal and write if . Two sets and are orthogonal and write if and are orthogonal for all and while is said to be orthogonal to an element if is orthogonal to . For , define the orthogonal or annihilator of to be .
Polar sets
Throughout, will be a pairing over . The absolute polar or polar of a subset of is the set:[4]
.
Dually, the absolute polar or polar of a subset of is denoted by and defined by
.
In this case, the absolute polar of a subset of is also called the absolute prepolar or prepolar of and may be denoted by .
The polar is necessarily a convex set containing where if is balanced then so is and if is a vector subspace of then so too is a vector subspace of .[5]
If then the bipolar of , denoted by , is the set . Similarly, if then the bipolar of is .
If is a vector subspace of , then and this is also equal to the real polar of .
Dual definitions and results
Given a pairing , define a new pairing where for all and all .[2] For the purpose of explaining the following convention, the map will be called the mirror of (this is not standard terminology).
There is a repeating theme in duality theory, which is that any definition for a pairing has a corresponding dual definition for the pairing .
Convention and Definition: Given any definition for a pairing , one obtains a dual definition by applying it to the pairing . This conventions also apply to theorems.
Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) for a pairing is given then this article will omit mention of the corresponding dual definition (or result) but nevertheless use it.
For instance, if " distinguishes points of " (resp, " is a total subset of ") is defined as above, then this convention immediately produces the dual definition of " distinguishes points of " (resp, " is a total subset of ").
This following notation is almost ubiquitous because it allows us to avoid having to assign a symbol to the mirror of .
Convention and Notation: If a definition and its notation for a pairing depends on the order of and (e.g. the definition of the Mackey topology on ) then by switching the order of and , then it is meant that definition applied to (e.g. actually denotes the topology ).
For instance, once the weak topology on is defined, which is denoted by , then this definition will automatically be applied to the pairing so as to obtain the definition of the weak topology on , where this topology will be denoted by rather than .
Identification of with
Although it is technically incorrect and an abuse of notation, this article will also adhere to the following nearly ubiquitous convention:
This article will use the common practice of treating a pairing interchangeably with and also denoting by .
Examples
Restriction of a pairing
Suppose that is a pairing, is a vector subspace of , and is a vector subspace of . Then the restriction of to is the pairing . If is a duality then it's possible for a restrictions to fail to be a duality (e.g. if and ).
This article will use the common practice of denoting the restriction by .
Canonical duality on a vector space
Suppose that is a vector space and let denote the algebraic dual space of (that is, the space of all linear functionals on ). There is a canonical duality where , which is called the evaluation map or the natural or canonical bilinear functional on . Note in particular that for any , is just another way of denoting ; i.e. .
If is a vector subspace of then the restriction of to is called the canonical pairing where if this pairing is a duality then it is instead called the canonical duality. Clearly, always distinguishes points of so the canonical pairing is a dual system if and only if separates points of . The following notation is now nearly ubiquitous in duality theory.
The evaluation map will be denoted by (rather than by ) and will be written rather than .
Assumption: As is common practice, if is a vector space and is a vector space of linear functionals on , then unless stated otherwise, it will be assumed that they are associated with the canonical pairing .
If is a vector subspace of then distinguishes points of (or equivalently, is a duality) if and only if distinguishes points of , or equivalently if is total (i.e. for all implies ).[2]
Canonical duality on a topological vector space
Suppose is a topological vector space (TVS) with continuous dual space. Then the restriction of the canonical duality to × defines a pairing for which separates points of . If separates points of (which is true if, for instance, is a Hausdorff locally convex space) then this pairing forms a duality.[3]
Assumption: As is commonly done, whenever is a TVS then, unless indicated otherwise, it will be assumed without comment that it's associated with the canonical pairing .
Polars and duals of TVSs
The following result shows that the continuous linear functionals on a TVS are exactly those linear functionals that are bounded on a neighborhood of the origin.
Theorem[2] — Let be a TVS with algebraic dual and let be a basis of neighborhoods of at the origin. Under the canonical duality , the continuous dual space of is the union of all as ranges over (where the polars are taken in ).
Inner product spaces and complex conjugate spaces
A pre-Hilbert space is a dual pairing if and only if is vector space over or has dimension . Here it is assumed that the sesquilinear form is conjugate homogeneous in its second coordinate and homogeneous in its first coordinate.
If is a complex Hilbert space then forms a dual system if and only if . If is non-trivial then doesn't even form pairing since the inner product is sesquilinear rather than bilinear.[2]
Suppose that is a complex pre-Hilbert space with multiplication denoted by juxtaposition or by a dot . Define the map
by ,
where the right hand side uses the scalar multiplication of . Let denote the complex conjugate vector space of , where denotes the additive group of (so vector addition in is identical to vector addition in ) but with scalar multiplication in being the map rather than the scalar multiplication that is endowed with.
The map defined by is linear in both coordinates[note 2] and so forms a dual pairing.
Other examples
Suppose , , and for all and . Then is a pairing such that distinguishes points of , but does not distinguish points of . Furthermore, .
Let , , (where is such that ), and . Then is a dual system.
Let and be vector spaces over the same field . Then the bilinear form places and in duality.[3]
A sequence space and its beta dual with the bilinear map defined as for , forms a dual system.
Weak topology
Suppose that is a pairing of vector spaces over . If then the weak topology on induced by (and ) is the weakest TVS topology on , denoted by or simply , making all maps continuous, as ranges over .[2] The notation , , or (if no confusion could arise) simply is used to denote endowed with the weak topology . If it is not indicated what the subset is, then by the weak topology on it is meant the weak topology on induced by . Similarly, if then the dual definition of the weak topology on induced by (and ), which is denoted by or simply (see footnote for details).[note 3] Importantly, the weak topology depends entirely on the function and the usual topology on (the topology doesn't even depend on the algebraic structures of and ).
Definition and Notation: If " " is attached to a topological definition (e.g. -converges, -bounded, , etc.) then it means that definition when the first space (i.e. ) carries the topology. Mention of or even and may be omitted if no confusion will arise. So for instance, if a sequence in " -converges" or "weakly converges" then this means that it converges in whereas if it were a sequence in then this would mean that it converges in ).
The topology is locally convex since it is determined by the family of seminorms defined by , as ranges over .[2] If and is a net in , then -converges to if converges to in .[2] A net -converges to if and only if for all , converges to . If is a sequence of orthonormal vectors in Hilbert space, then converges weakly to 0 but does not norm-converge to 0 (or any other vector).[2]
If is a pairing and is a proper vector subspace of such that is a dual pair, then is strictly coarser than .[2]
Bounded subsets
A subset of is -bounded if and only if for all , where .
Hausdorffness
If is a pairing then the following are equivalent:
distinguishes points of ;
The map defines an injection from into the algebraic dual space of ;[2]
The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of .
Weak representation theorem[2] — Let be a pairing over the field . Then the continuous dual space of is Furthermore,
If is a continuous linear functional on then there exists some such that ; if such a exists then it is unique if and only if distinguishes points of .
The continuous dual space of may be identified with , where .
This is true regardless of whether or not distinguishes points of or distinguishes points of .
Orthogonals, quotients, and subspaces
If is a pairing then for any subset of :
and this set is -closed;[2]
;[2]
Thus if is a -closed vector subspace of then .
If is a family of -closed vector subspaces of then
. [2]
If is a family of subsets of then .[2]
If is a normed space then under the canonical duality, is norm closed in and is norm closed in .[2]
Subspaces
Suppose that is a vector subspace of and let denote the restriction of to . The weak topology on is identical to the subspace topology that inherits from .
Also, is a paired space (where means ) where is defined by
.
The topology is equal to the subspace topology that inherits from .[6] Furthermore, if is a dual system then so is .[6]
Quotients
Suppose that is a vector subspace of . Then is a paired space where is defined by
.
The topology is identical to the usual quotient topology induced by on .[6]
Polars and the weak topology
If is a locally convex space and if is a subset of the continuous dual space , then is -bounded if and only if for some barrel in .[2]
The following results are important for defining polar topologies.
If is a pairing and , then:[2]
The polar of , , is a closed subset of .
The polars of the following sets are identical: (a) ; (b) the convex hull of ; (c) the balanced hull of ; (d) the -closure of ; (e) the -closure of the convex balanced hull of .
The bipolar theorem: The bipolar of , , is equal to the -closure of the convex balanced hull of .
The bipolar theorem in particular "is an indispensable tool in working with dualities."[5]
is -bounded if and only if is absorbing in .
If in addition distinguishes points of then is -bounded if and only if it is -totally bounded.
If is a pairing and is a locally convex topology on that is consistent with duality, then a subset of is a barrel in if and only if is the polar of some -bounded subset of .[7]
Transposes
Transpose of a linear map with respect to pairings
Let and be pairings over and let be a linear map.
For all , let be the map defined by . It is said that 's transpose or adjoint is well-defined if the following conditions are satisfies:
distinguishes points of (or equivalently, the map from into the algebraic dual
is injective), and
, where .
In this case, for any there exists (by condition 2) a unique (by condition 1) such that ), where this element of will be denoted by . This defines a linear map
called the transpose of adjoint of with respect to and (this should not to be confused with the Hermitian adjoint). It is easy to see that the two conditions mentioned above (i.e. for "the transpose is well-defined") are also necessary for to be well-defined. For every , the defining condition for is
,
that is,
for all .
By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form ,[note 4],[note 5],[note 6],[note 7] etc. (see footnote for details).
Properties of the transpose
Throughout, and be pairings over and will be a linear map whose transpose is well-defined.
is injective (i.e. ) if and only if the range of is dense in .[2]
If in addition to being well-defined, the transpose of is also well-defined then .
Suppose is a pairing over and is a linear map whose transpose is well-defined. Then the transpose of , which is , is well-defined and .
If is a vector space isomorphism then is bijective, the transpose of , which is , is well-defined, and [2]
Let and let denotes the absolute polar of , then:[2]
;
if for some , then ;
if is such that , then ;
if and are weakly closed disks then if and only if ;
.
These results hold when the real polar is used in place of the absolute polar.
If and are normed spaces under their canonical dualities and if is a continuous linear map, then .[2]
Weak continuity
A linear map is weakly continuous (with respect to and ) if is continuous.
The following result shows that the existence of the transpose map is intimately tied to the weak topology.
Proposition — Assume that distinguishes points of and is a linear map. Then the following are equivalent:
is weakly continuous (i.e. is continuous);
;
the transpose of is well-defined.
If is weakly continuous then
is weakly continuous (i.e. is continuous);
the transpose of is well-defined if and only if distinguishes points of , in which case .
Weak topology and the canonical duality
Suppose that is a vector space and that is its the algebraic dual. Then every -bounded subset of is contained in a finite dimensional vector subspace and every vector subspace of is -closed.[2]
Weak completeness
Call -complete or (if no ambiguity can arise) weakly-complete if is a complete vector space. There exist Banach spaces that are not weakly-complete (despite being complete).[2]
If is a vector space then under the canonical duality, is complete.[2] Conversely, if is a Hausdorff locally convex TVS with continuous dual space , then is complete if and only if (i.e. the map defined by sending to the evaluation map at (i.e. ) is a bijection).[2]
In particular, if is a vector subspace of such that separates points of , then is complete if and only if .
Identification of Y with a subspace of the algebraic dual
If distinguishes points of and if denotes the range of the injection then is a vector subspace of the algebraic dual space of and the pairing becomes canonically identified with the canonical pairing (where is the natural evaluation map). In particular, in this situation it will be assumed without loss of generality that is a vector subspace of 's algebraic dual and is the evaluation map.
Convention: Often, whenever is injective (especially when forms a dual pair) then it is common practice to assume without loss of generality that is a vector subspace of the algebraic dual space of , that is the natural evaluation map, and also denote by .
In a completely analogous manner, if distinguishes points of then it is possible for to be identified as a vector subspace of 's algebraic dual space.[3]
Algebraic adjoint
In the spacial case where the dualities are the canonical dualities and , the transpose of a linear map is always well-defined. This transpose is called the algebraic adjoint of and it will be denoted by ; that is, . In this case, for all , [2][8] where the defining condition for is:
for all ,
or equivalently, for all .
Examples
If for some integer , is a basis for with dual basis , is a linear operator, and the matrix representation of with respect to is , then the transpose of is the matrix representation with respect to of .
Weak continuity and openness
Suppose that and are canonical pairings (so and ) that are dual systems and let be a linear map. Then is weakly continuous if and only if it satisfies any of the following equivalent conditions:[2]
is continuous;
the transpose of F, , with respect to and is well-defined.
If is weakly continuous then will be continuous and furthermore, [8]
A map between topological spaces is relatively open if is an open mapping, where is the range of .[2]
Suppose that and are dual systems and is a weakly continuous linear map. Then the following are equivalent:[2]
is relatively open;
The range of is -closed in ;
Furthermore,
is injective (resp. bijective) if and only if is surjective (resp. bijective);
is surjective if and only if is relatively open and injective.
Transpose of a map between TVSs
The transpose of map between two TVSs is defined if and only if is weakly continuous.
If is a linear map between two Hausdorff locally convex topological vector spaces then:[2]
If is continuous then it is weakly continuous and is both Mackey continuous and strongly continuous.
If is weakly continuous then it is both Mackey continuous and strongly continuous (defined below).
If is weakly continuous then it is continuous if and only if maps equicontinuous subsets of to equicontinuous subsets of .
If and are normed spaces then is continuous if and only if it is weakly continuous, in which case .
If is continuous then is relatively open if and only if is weakly relatively open (i.e. is relatively open) and every equicontinuous subsets of is the image of some equicontinuous subsets of .
If is continuous injection then is a TVS-embedding (or equivalently, a topological embedding) if and only if every equicontinuous subsets of is the image of some equicontinuous subsets of .
Metrizability and separability
Let be a locally convex space with continuous dual space and let .[2]
If is equicontinuous or -compact, and if is such that is dense in , then the subspace topology that inherits from is identical to the subspace topology that inherits from .
If is separable and is equicontinuous then , when endowed with the subspace topology induced by , is metrizable.
If is separable and metrizable, then is separable.
If is a normed space then is separable if and only if the closed unit call the continuous dual space of is metrizable when given the subspace topology induced by .
If is a normed space whose continuous dual space is separable (when given the usual norm topology), then is separable.
Polar topologies and topologies compatible with pairing
Starting with only the weak topology, the use of polar sets produces a range of locally convex topologies. Such topologies are called polar topologies. The weak topology is the weakest topology of this range.
Throughout, will be a pairing over and will be a non-empty collection of -bounded subsets of .
Polar topologies
The polar topology on determined by (and ) or the -topology on is the unique topological vector space (TVS) topology on for which
forms a subbasis of neighborhoods at the origin.[2] When is endowed with this -topology then it is denoted by Y. Every polar topology is necessarily locally convex.[2] When is a directed set with respect to subset inclusion (i.e. if for all there exists some such that ) then this neighborhood subbasis at 0 actually forms a neighborhood basis at 0.[2]
The following table lists some of the more important polar topologies.
Notation: If denotes a polar topology on then endowed with this topology will be denoted by , or simply (e.g. for we'd have so that , and all denote with endowed with ).
("topology of uniform convergence on ...")
Notation
Name ("topology of...")
Alternative name
finite subsets of (or -closed disked hulls of finite subsets of )
pointwise/simple convergence
weak/weak* topology
-compact disks
Mackey topology
-compact convex subsets
compact convex convergence
-compact subsets (or balanced -compact subsets)
compact convergence
-bounded subsets
bounded convergence
strong topology Strongest polar topology
Definitions involving polar topologies
Continuity
A linear map is Mackey continuous (with respect to and ) if is continuous.[2]
A linear map is strongly continuous (with respect to and ) if is continuous.[2]
Bounded subsets
A subset of is weakly bounded (resp. Mackey bounded, strongly bounded) if it is bounded in (resp. bounded in , bounded in ).
Topologies compatible with a pair
If is a pairing over and is a vector topology on then is a topology of the pairing and that it is compatible (or consistent) with the pairing if it is locally convex and if the continuous dual space of .[note 8] If distinguishes points of then by identifying as a vector subspace of 's algebraic dual, the defining condition becomes: .[2] Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff,[3][9] which it would have to be if distinguishes the points of (which these authors assume).
The weak topology is compatible with the pairing (as was shown in the Weak representation theorem) and it is in fact the weakest such topology. There is a strongest topology compatible with this pairing and that is the Mackey topology. If is a normed space that is not reflexive then the usual norm topology on its continuous dual space is not compatible with the duality .[2]
Mackey-Arens theorem
The following is one of the most important theorems in duality theory.
Mackey-Arens theorem I[2] — Let will be a pairing such that distinguishes the points of and let be a locally convex topology on (not necessarily Hausdorff). Then is compatible with the pairing if and only if is a polar topology determined by some collection of -compact disks that cover[note 9].
It follows that the Mackey topology , which recall is the polar topology generated by all -compact disks in , is the strongest locally convex topology on that is compatible with the pairing . A locally convex space whose given topology is identical to the Mackey topology is called a Mackey space. The following consequence of the above Mackey-Arens theorem is also called the Mackey-Arens theorem.
Mackey-Arens theorem II[2] — Let will be a pairing such that distinguishes the points of and let be a locally convex topology on . Then is compatible with the pairing if and only if .
Mackey's theorem, barrels, and closed convex sets
If is a TVS (over or ) then a half-space is a set of the form for some real and some continuous real linear functional on .
Theorem — If is a locally convex space (over or ) and if is a non-empty closed and convex subset of , then is equal to the intersection of all closed half spaces containing it.[10]
The above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality; that is, if and are any locally convex topologies on with the same continuous dual spaces, then a convex subset of is closed in the topology if and only if it is closed in the topology. This implies that the -closure of any convex subset of is equal to its -closure and that for any -closed disk in , .[2] In particular, if is a subset of then is a barrel in if and only if it is a barrel in .[2]
The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets.
Theorem[2] — Let will be a pairing such that distinguishes the points of and let be a topology of the pair. Then a subset of is a barrel in if and only if it is equal to the polar of some -bounded subset of .
If is a topological vector space then:[2][11]
A closed absorbing and balanced subset of absorbs each convex compact subset of (i.e. there exists a real such that contains that set).
If is Hausdorff and locally convex then every barrel in absorbs every convex bounded complete subset of .
All of this leads to Mackey's theorem, which is one of the central theorems in the theory of dual systems. In short, it states the bounded subsets are the same for any two Hausdorff locally convex topologies that are compatible with the same duality.
Mackey's theorem[11][2] — Suppose that is a Hausdorff locally convex space with continuous dual space and consider the canonical duality . If is any topology on that is compatible with the duality on then the bounded subsets of are the same as the bounded subsets of .
Examples
Space of finite sequences
Let denote the space of all sequences of scalars such that for all sufficiently large . Let and define a bilinear map by
.
Then .[2] Moreover, a subset is -bounded (resp. -bounded) if and only if there exists a sequence of positive real numbers such that for all and all indices (resp. and ).[2] It follows that there are weakly bounded (i.e. -bounded) subsets of that are not strongly bounded (i.e. not -bounded).
See also
Dual space – Vector space of linear functions of vectors returning scalars; generalizing the dot product
Dual topology
Duality (mathematics)
Inner product
L-semi-inner product – Generalization of inner products that applies to all normed spaces
Pairing
Polar set – Subset of all points that is bounded by some given point of a dual (in a dual pairing)
Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
Reductive dual pair
Strong dual space – Continuous dual space endowed with the topology of uniform convergence on bounded sets
Strong topology (polar topology) – Dual space topology of uniform convergence on bounded subsets
Topologies on spaces of linear maps
Weak topology – Topology where convergence of points is defined by the convergence of their image under continuous linear functionals
Notes
^A subset of is called total if for all , for all implies .
^That is linear in its first coordinate is obvious. Suppose is a scalar. Then , which shows that is linear in its second coordinate.
^The weak topology on is the weakest TVS topology on making all maps continuous, as ranges over . We also use the dual notation of , , or simply to denote endowed with the weak topology . If it is not indicated what the subset is, then by the weak topology on it is meant the weak topology on induced by .
^If is a linear map then 's transpose, , is well-defined if and only if distinguishes points of and . In this case, for each , the defining condition for is: .
^If is a linear map then 's transpose, , is well-defined if and only if distinguishes points of and . In this case, for each , the defining condition for is: .
^If is a linear map then 's transpose, , is well-defined if and only if distinguishes points of and . In this case, for each , the defining condition for is: .
^If is a linear map then 's transpose, , is well-defined if and only if distinguishes points of and . In this case, for each , the defining condition for is: .
^Of course, there is an analogous definition for topologies on to be "compatible it a pairing" but this article will only deal with topologies on .
^Recall that a collection of subsets of a set is said to cover if every point of is contained in some set belonging to the collection.
References
^Schaefer & Wolff 1999, p. 122.
^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ayNarici & Beckenstein 2011, pp. 225-273.
^ a b c d e fSchaefer & Wolff 1999, pp. 122–128.
^Trèves 2006, p. 195.
^ a bSchaefer & Wolff 1999, pp. 123–128.
^ a b cNarici & Beckenstein 2011, pp. 260-264.
^Narici & Beckenstein 2011, pp. 251-253.
^ a bSchaefer & Wolff 1999, pp. 128–130.
^Trèves 2006, pp. 368–377.
^Narici & Beckenstein 2011, p. 200.
^ a bTrèves 2006, pp. 371–372.
Bibliography
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Schmitt, Lothar M (1992). "An Equivariant Version of the Hahn–Banach Theorem". Houston J. Of Math. 18: 429–447.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.