Absorbing set

In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set.

Definition[]

Suppose that ${\displaystyle X}$ is a vector space over the field ${\displaystyle \mathbb {K} }$ of real numbers ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} .}$

Notation[]

Products of scalars and vectors

For any ${\displaystyle -\infty \leq r\leq R\leq \infty ,}$ vector ${\displaystyle x,}$ and subset ${\displaystyle A\subseteq X,}$ let

denote the open ball (respectively, the closed ball) of radius ${\displaystyle r}$ in ${\displaystyle \mathbb {K} }$ centered at ${\displaystyle 0,}$ and let

Similarly, if ${\displaystyle K\subseteq \mathbb {K} }$ and ${\displaystyle k}$ is a scalar then let ${\displaystyle KA=\{ka:k\in K,a\in A\},}$ ${\displaystyle Kx=\{kx:k\in K\},}$ ${\displaystyle kA=\{ka:a\in A\},}$ and ${\displaystyle \mathbb {K} x=\{kx:k\in \mathbb {K} \}=\operatorname {span} \{x\}.}$

One set absorbing another[]

If ${\displaystyle S}$ and ${\displaystyle A}$ are subsets of ${\displaystyle X,}$ then ${\displaystyle A}$ is said to absorb ${\displaystyle S}$ if it satisfies any of the following equivalent conditions:

1. Definition: There exists a real ${\displaystyle r>0}$ such that ${\displaystyle S\subseteq cA}$ for every scalar ${\displaystyle c}$ satisfying ${\displaystyle |c|\geq r.}$
   • If the scalar field is ${\displaystyle \mathbb {R} }$ then intuitively, "${\displaystyle A}$ absorbs ${\displaystyle S}$" means that if ${\displaystyle A}$ is perpetually "scaled up" or "inflated" (referring to ${\displaystyle tA}$ as ${\displaystyle t\to \infty }$) then eventually, all ${\displaystyle tA}$ will contain ${\displaystyle S}$ (for all positive ${\displaystyle t>0}$ sufficiently large); and similarly, ${\displaystyle tA}$ must also eventually contain ${\displaystyle S}$ for all negative ${\displaystyle t<0}$ sufficiently large in magnitude.
   • This definition depends on the canonical norm on the underlying scalar field, which ties this definition to the usual Euclidean topology on the scalar field. Consequently, the definition of an absorbing set (given below) is also tied to this topology.
2. There exists a real ${\displaystyle r>0}$ such that ${\displaystyle cS\subseteq A}$ for every scalar ${\displaystyle c\neq 0}$ satisfying ${\displaystyle |c|\leq r.}$
   • If it is known that ${\displaystyle 0\in A}$ then the restriction ${\displaystyle c\neq 0}$ may be removed, giving the characterization: There exists a real ${\displaystyle r>0}$ such that ${\displaystyle cS\subseteq A}$ for every scalar ${\displaystyle c}$ satisfying ${\displaystyle |c|\leq r.}$
3. There exists a real ${\displaystyle r>0}$ such that ${\displaystyle \left(B_{r}\setminus \{0\}\right)S\subseteq A.}$
   • The closed ball ${\displaystyle B_{\leq r}}$ (with the origin removed) can be used in place of the open ball ${\displaystyle B_{r},}$ giving the next characterization.
4. There exists a real ${\displaystyle r>0}$ such that ${\displaystyle \left(B_{\leq r}\setminus \{0\}\right)S\subseteq A.}$

If ${\displaystyle A}$ is a balanced set then to this list can be appended:

5. There exists a scalar ${\displaystyle c\neq 0}$ such that ${\displaystyle S\subseteq cA.}$
6. There exists a scalar ${\displaystyle c\neq 0}$ such that ${\displaystyle cS\subseteq A.}$

A set is said to absorb a point ${\displaystyle x}$ if it absorbs the singleton set ${\displaystyle \{x\}.}$ A set ${\displaystyle A}$ absorbs the origin if and only if it contains the origin; that is, if and only if ${\displaystyle 0\in A.}$ Every set absorbs the empty set.

Examples: Suppose that ${\displaystyle X}$ is equal to either ${\displaystyle \mathbb {R} ^{2}}$ or ${\displaystyle \mathbb {C} .}$ If ${\displaystyle A:=S^{1}\cup \{\mathbf {0} \}}$ is the unit circle (centered at the origin) together with the origin, then ${\displaystyle \{\mathbf {0} \}}$ is the one and only non-empty set that ${\displaystyle A}$ absorbs. Moreover, there does not exist any non-empty subset of ${\displaystyle X}$ that is absorbed by unit circle ${\displaystyle S^{1}.}$ In contrast, every neighborhood of the origin absorbs every bounded subset of ${\displaystyle X}$ (and so in particular, absorbs every singleton subset/point).

Absorbing set[]

A subset ${\displaystyle A}$ of a vector space ${\displaystyle X}$ over a field ${\displaystyle \mathbb {K} }$ is called an absorbing (or absorbent) subset of ${\displaystyle X}$ and is said to be absorbing in ${\displaystyle X}$ if it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one, starting with the definition):

1. Definition: For every ${\displaystyle x\in X,A}$ absorbs ${\displaystyle \{x\}.}$ Said differently, ${\displaystyle A}$ absorbs every point of ${\displaystyle X.}$
   • So in particular, ${\displaystyle A}$ can not be absorbing if ${\displaystyle 0\not \in A.}$
2. For every ${\displaystyle x\in X,}$ there exists a real ${\displaystyle r>0}$ such that ${\displaystyle x\in cA}$ for any scalar ${\displaystyle c\in \mathbb {K} }$ satisfying ${\displaystyle |c|\geq r.}$
3. For every ${\displaystyle x\in X,}$ there exists a real ${\displaystyle r>0}$ such that ${\displaystyle cx\in A}$ for any scalar ${\displaystyle c\in \mathbb {K} }$ satisfying ${\displaystyle |c|\leq r.}$
4. For every ${\displaystyle x\in X,}$ there exists a real ${\displaystyle r>0}$ such that ${\displaystyle B_{r}x\subseteq A.}$
   • Here ${\displaystyle B_{r}=\{c\in \mathbb {K} :|c|<r\}}$ is the open ball of radius ${\displaystyle r}$ in the scalar field centered at the origin and ${\displaystyle B_{r}x=\left\{cx:c\in B_{r}\right\}=\{cx:c\in \mathbb {K} {\text{ and }}|c|<r\}.}$
   • The closed ball can be used in place of the open ball.
5. For every ${\displaystyle x\in X,}$ there exists a real ${\displaystyle r>0}$ such that ${\displaystyle B_{r}x\subseteq A\cap \mathbb {K} x,}$ where ${\displaystyle \mathbb {K} x=\operatorname {span} \{x\}.}$
   • Proof: This follows from the previous condition since ${\displaystyle B_{r}x\subseteq \mathbb {K} x,}$ so that ${\displaystyle B_{r}x\subseteq A}$ if and only if ${\displaystyle B_{r}x\subseteq A\cap \mathbb {K} x.}$
   • Connection to topology: If ${\displaystyle \mathbb {K} x}$ is given its usual Hausdorff Euclidean topology then the set ${\displaystyle B_{r}x}$ is a neighborhood of the origin in ${\displaystyle \mathbb {K} x;}$ thus, there exists a real ${\displaystyle r>0}$ such that ${\displaystyle B_{r}x\subseteq A\cap \mathbb {K} x}$ if and only if ${\displaystyle A\cap \mathbb {K} x}$ is a neighborhood of the origin in ${\displaystyle \mathbb {K} x.}$
   • Every 1-dimensional vector subspace of ${\displaystyle X}$ is of the form ${\displaystyle \mathbb {K} x=\operatorname {span} \{x\}}$ for some non-zero ${\displaystyle x\in X}$ and if this 1-dimensional space ${\displaystyle \mathbb {K} x}$ is endowed with the unique Hausdorff vector topology, then the map ${\displaystyle \mathbb {K} \to \mathbb {K} x}$ defined by ${\displaystyle c\mapsto cx}$ is necessarily a TVS-isomorphism (where as usual, ${\displaystyle \mathbb {K} }$ has the normed Euclidean topology).
6. ${\displaystyle A}$ contains the origin and for every 1-dimensional vector subspace ${\displaystyle Y}$ of ${\displaystyle X,}$ ${\displaystyle A\cap Y}$ is a neighborhood of the origin in ${\displaystyle Y}$ when ${\displaystyle Y}$ is given its unique Hausdorff vector topology.
   • The Hausdorff vector topology on a 1-dimensional vector space is necessarily TVS-isomorphic to ${\displaystyle \mathbb {K} }$ with its usual normed Euclidean topology.
   • Intuition: This condition shows that it is only natural that any neighborhood of 0 in any topological vector space (TVS) ${\displaystyle X}$ be absorbing: if ${\displaystyle U}$ is a neighborhood of the origin in ${\displaystyle X}$ then it would be pathological if there existed any 1-dimensional vector subspace ${\displaystyle Y}$ in which ${\displaystyle U\cap Y}$ was not a neighborhood of the origin in at least some TVS topology on ${\displaystyle Y.}$ The only TVS topologies on ${\displaystyle Y}$ are the Hausdorff Euclidean topology and the trivial topology, which is a subset of the Euclidean topology. Consequently, it is natural to expect for ${\displaystyle U\cap Y}$ to be a neighborhood of ${\displaystyle 0}$ in the Euclidean topology for all 1-dimensional vector subspaces ${\displaystyle Y,}$ which is exactly the condition that ${\displaystyle U}$ be absorbing in ${\displaystyle X.}$ The fact that all neighborhoods of the origin in all TVSs are necessarily absorbing means that this pathological behavior does not occur. The reason why the Euclidean topology is distinguished is ultimately due to the defining requirement on TVS topologies that scalar multiplication ${\displaystyle \mathbb {K} \times X\to X}$ be continuous when the scalar field ${\displaystyle \mathbb {K} }$ is given the Euclidean topology.
   • This condition is equivalent to: For every ${\displaystyle x\in X,}$ ${\displaystyle A\cap \operatorname {span} \{x\}}$ is a neighborhood of ${\displaystyle 0}$ in ${\displaystyle \operatorname {span} \{x\}=\mathbb {K} x}$ when ${\displaystyle \operatorname {span} \{x\}}$ is given its unique Hausdorff TVS topology.
7. ${\displaystyle A}$ contains the origin and for every 1-dimensional vector subspace ${\displaystyle Y}$ of ${\displaystyle X,}$ ${\displaystyle A\cap Y}$ is absorbing in the ${\displaystyle Y.}$
   • Here "absorbing" means absorbing according to any defining condition other than this one.
   • This shows that the property of being absorbing in ${\displaystyle X}$ depends only on how ${\displaystyle A}$ behaves with respect to 1 (or 0) dimensional vector subspaces of ${\displaystyle X.}$ In contrast, if a finite-dimensional vector subspace ${\displaystyle Z}$ of ${\displaystyle X}$ has dimension ${\displaystyle n>1}$ then ${\displaystyle A\cap Z}$ being absorbing in ${\displaystyle Z}$ is no longer sufficient to guarantee that ${\displaystyle A\cap Z}$ is a neighborhood of the origin in ${\displaystyle Z}$ when ${\displaystyle Z}$ is endowed with its unique Hausdorff TVS topology (although it will still be a necessary condition). For this to happen, it suffices for ${\displaystyle A\cap Z}$ to be a barrel in this Hausdorff TVS ${\displaystyle Z}$ (because every finite-dimensional Euclidean space is a barrelled space).

If ${\displaystyle \mathbb {K} =\mathbb {R} }$ then to this list can be appended:

8. The algebraic interior of ${\displaystyle A}$ contains the origin (that is, ${\displaystyle 0\in {}^{i}A}$).

If ${\displaystyle A}$ is balanced then to this list can be appended:

9. For every ${\displaystyle x\in X,}$ there exists a scalar ${\displaystyle c\neq 0}$ such that ${\displaystyle x\in cA.}$^[1]

If ${\displaystyle A}$ is convex or balanced then to this list can be appended:

10. For every ${\displaystyle x\in X,}$ there exists a positive real ${\displaystyle r>0}$ such that ${\displaystyle rx\in A.}$
    • The proof that a balanced set ${\displaystyle A}$ satisfying this condition is necessarily absorbing in ${\displaystyle X}$ is almost immediate from the definition of a "balanced set".
    • The proof that a convex set ${\displaystyle A}$ satisfying this condition is necessarily absorbing in ${\displaystyle X}$ is less trivial (but not difficult). A detailed proof is given in this footnote^{[proof 1]} and a summary is given below.
      • Summary of proof: By assumption, for any non-zero ${\displaystyle 0\neq y\in X,}$ it is possible to pick positive real ${\displaystyle r>0}$ and ${\displaystyle R>0}$ such that ${\displaystyle Ry\in A}$ and ${\displaystyle r(-y)\in A}$ so that the convex set ${\displaystyle A\cap \mathbb {R} y}$ contains the open sub-interval ${\displaystyle (-r,R)y:=\{ty:-r<t<R,t\in \mathbb {R} \},}$ which contains the origin (the set ${\displaystyle A\cap \mathbb {R} y}$ is also called an interval because every non-empty convex subset of ${\displaystyle \mathbb {R} }$ is an interval). Give ${\displaystyle \mathbb {K} y}$ its unique Hausdorff vector topology so it remains to show that ${\displaystyle A\cap \mathbb {K} y}$ is a neighborhood of the origin in ${\displaystyle \mathbb {K} y.}$ If ${\displaystyle \mathbb {K} =\mathbb {R} }$ then we are done, so assume that ${\displaystyle \mathbb {K} =\mathbb {C} .}$ The set ${\displaystyle S\,:=\,(A\cap \mathbb {R} y)\,\cup \,(A\cap \mathbb {R} (iy))\,\subseteq \,A\cap (\mathbb {C} y)}$ is a union of two intervals, each of which contains an open sub-interval that contains the origin; moreover, the intersection of these two intervals is precisely the origin. So the convex hull of ${\displaystyle S,}$ which is contained in the convex set ${\displaystyle A\cap \mathbb {C} y,}$ clearly contains an open ball around the origin. ${\displaystyle \blacksquare }$
11. For every ${\displaystyle x\in X,}$ there exists a positive real ${\displaystyle r>0}$ such that ${\displaystyle x\in rA.}$
    • This condition is equivalent to: every ${\displaystyle x\in X}$ belongs to the set ${\displaystyle \bigcup _{0<r<\infty }rA=\{ra:0<r<\infty ,a\in A\}=(0,\infty )A.}$ This happens if and only if ${\displaystyle X=(0,\infty )A,}$ which gives the next characterization.
12. ${\displaystyle (0,\infty )A=X.}$
    • It can be shown that for any subset ${\displaystyle T}$ of ${\displaystyle X,}$ ${\displaystyle (0,\infty )T=X}$ if and only if ${\displaystyle T\cap (0,\infty )x\neq \varnothing {\text{ for every }}x\in X.}$
13. For every ${\displaystyle x\in X,A\cap (0,\infty )x\neq \varnothing ,}$ where ${\displaystyle (0,\infty )x:=\{rx:0<r<\infty \}}$

If ${\displaystyle 0\in A}$ (which is necessary for ${\displaystyle A}$ to be absorbing) then it suffices to check any of the above conditions for all non-zero ${\displaystyle x\in X,}$ rather than all ${\displaystyle x\in X.}$

Examples and sufficient conditions[]

For one set to absorb another[]

Let ${\displaystyle F:X\to Y}$ be a linear map between vector spaces and let ${\displaystyle B\subseteq X}$ and ${\displaystyle C\subseteq Y}$ be balanced sets. Then ${\displaystyle C}$ absorbs ${\displaystyle F(B)}$ if and only if ${\displaystyle F^{-1}(C)}$ absorbs ${\displaystyle B.}$^[2]

If a set ${\displaystyle A}$ absorbs another set ${\displaystyle B}$ then any superset of ${\displaystyle A}$ also absorbs ${\displaystyle B.}$ A set ${\displaystyle A}$ absorbs the origin if and only if the origin is an element of ${\displaystyle A.}$

For a set to be absorbing[]

In a semi normed vector space the unit ball is absorbing. More generally, if ${\displaystyle X}$ is a topological vector space (TVS) then any neighborhood of the origin in ${\displaystyle X}$ is absorbing in ${\displaystyle X.}$ This fact is one of the primary motivations for even defining the property "absorbing in ${\displaystyle X.}$"

If ${\displaystyle D\neq \varnothing }$ is a disk in ${\displaystyle X}$ then ${\displaystyle \operatorname {span} D=\bigcup _{n=1}^{\infty }nD}$ so that in particular, ${\displaystyle D}$ is an absorbing subset of ${\displaystyle \operatorname {span} D.}$^[3] Thus if ${\displaystyle D}$ is a disk in ${\displaystyle X,}$ then ${\displaystyle X}$ is absorbing in ${\displaystyle X}$ if and only if ${\displaystyle \operatorname {span} D=X.}$

Any superset of an absorbing set is absorbing. Thus the union of any family of (one or more) absorbing sets is absorbing. The intersection of a finite family of (one or more) absorbing sets is absorbing.

The image of an absorbing set under a surjective linear operator is again absorbing. The inverse image of an absorbing subset (of the codomain) under a linear operator is again absorbing (in the domain).

Properties[]

Every absorbing set contains the origin.

If ${\displaystyle D}$ is an absorbing disk in a vector space ${\displaystyle X}$ then there exists an absorbing disk ${\displaystyle E}$ in ${\displaystyle X}$ such that ${\displaystyle E+E\subseteq D.}$^[4]

See also[]

• Algebraic interior – Generalization of topological interior
• Absolutely convex set
• Balanced set – Construct in functional analysis
• Bornivorous set – A set that can absorb any bounded subset
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set that intersects every line into a single line segment
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Radial set
• Star domain
• Symmetric set
• Topological vector space – Vector space with a notion of nearness

Notes[]

Citations[]

References[]

• Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. Vol. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
• Nicolas, Bourbaki (2003). Topological vector spaces Chapter 1-5 (English Translation). New York: Springer-Verlag. p. I.7. ISBN 3-540-42338-9.
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Diestel, Joe (2008). The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. Vol. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
• Dineen, Seán (1981). Complex Analysis in Locally Convex Spaces. North-Holland Mathematics Studies. Vol. 57. Amsterdam New York New York: North-Holland Pub. Co., Elsevier Science Pub. Co. ISBN 978-0-08-087168-4. OCLC 16549589.
• Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. . Vol. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. OCLC 316549583.
• ; (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. Vol. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Keller, Hans (1974). Differential Calculus in Locally Convex Spaces. Lecture Notes in Mathematics. Vol. 417. Berlin New York: Springer-Verlag. ISBN 978-3-540-06962-1. OCLC 1103033.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: . ISBN 978-0-387-05644-9. OCLC 539541.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. . Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. p. 4.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Thompson, Anthony C. (1996). Minkowski Geometry. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. ISBN 0-521-40472-X.
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. Vol. 3. New York: Springer-Verlag. p. 11. ISBN 0-387-98726-6.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Schaefer, H. H. (1999). Topological Vector Spaces. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
• (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.