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 X is a vector space over the field 𝕂 of real or complex numbers .

Notation

Products of scalars and vectors

Notation: Let A be a subset of X, xX, K ⊆ 𝕂 a set of scalars, k0 ∈ 𝕂 a scalar, and -∞ ≤ tT ≤ ∞. Define:

  • K A := { k a : kK, aA}
    • k0 A := { k0 a : aA}.
    • K x := { k x : kK}.
  • 𝕂 x := { k x : k ∈ 𝕂} = span { x}
  • x := { r x : r ∈ ℝ}
  • (t, T) x := { r x : t < r < T} and (t, T) A := { r a : t < r < T, aA}
Balls in 𝕂

Notation: If r > 0 then denote the open ball of radius r centered at the origin in 𝕂 (where 𝕂 is or ) by

Br := B𝕂
r
:= { c ∈ 𝕂 : |c| < r
}

and denote the closed ball of radius r > 0 centered at the origin in 𝕂 by

Br := B𝕂
r
:= { c ∈ 𝕂 : |c|r
}.

One set absorbing another

Definition: If S and A are subsets of X, we say that A absorbs S if it satisfies any of the following equivalent conditions:

  1. There exists a real r > 0 such that Sc A for every scalar c satisfying |c|r;
    • If the scalar field is then intuitively, "A absorbs S" means that if A is perpetually "scaled up" or "inflated" (i.e. t A with t → ∞) then eventually all t A will contain S.
    • Observe that 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.
  2. There exists a real r > 0 such that c SA for every scalar c ≠ 0 satisfying |c|r;
    • If it is known that 0 ∈ A then we may remove the restriction c ≠ 0.
  3. There exists a real r > 0 such that (Br ∖ { 0 }) SA.
    • The closed ball (with the origin removed) can be used in place of the open ball.

If A is balanced then we can add to this list:

  1. There exists a scalar c ≠ 0 such that Sc A;
  2. There exists a scalar c ≠ 0 such that c SA.

Absorbing set

Definition: A subset A of a vector space X over a field 𝕂 is called absorbing or absorbent in X if it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one):

  1. For every xX, A absorbs { x }.
  2. For every xX, there exists a real r > 0 such that xc A for any scalar c ∈ 𝕂 satisfying |c|r.
  3. For every xX, there exists a real r > 0 such that c xA for any scalar c ∈ 𝕂 satisfying |c|r.
  4. For every xX, there exists a real r > 0 such that Br xA.
    • Here Br := B𝕂
      r
      := { c ∈ 𝕂 : |c| < r }
      is the open ball of radius r in 𝕂 centered at the origin and Br x := { c x : c ∈ B𝕂
      r
      } = { c x : c ∈ 𝕂 and |c| < r
      }.
    • The closed ball can be used in place of the open ball.
  5. For every xX, there exists a real r > 0 such that Br xA ∩ 𝕂 x, where 𝕂 x = span { x}.
    • Proof: This follows from the previous condition since Br x ⊆ 𝕂 x, so that Br xA if and only if Br xA ∩ 𝕂 x.
    • Connection to topology: Note that if 𝕂 x is given its usual Hausdorff Euclidean topology then the set Br x is a neighborhood of the origin in 𝕂 x; thus, there exists a real r > 0 such that Br xA ∩ 𝕂 x if and only if A ∩ 𝕂 x is a neighborhood of the origin in 𝕂 x.
    • Note that every 1-dimensional vector subspace of X is of the form 𝕂 x = span { x} for some non-zero xX and that if the 1-dimensional space 𝕂 x is endowed with the unique Hausdorff vector topology, then the map 𝕂 → 𝕂 x given by cc x is a TVS-isomorphism (where as usual, 𝕂 has the normed Euclidean topology).
  6. A contains the origin and for every 1-dimensional vector subspace Y of X, AY is a neighborhood of the origin in Y when Y is given its unique Hausdorff vector topology.
    • The Hausdorff vector topology on a 1-dimensional vector space is necessarily TVS-isomorphic to 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) X be absorbing: if U is a neighborhood of the origin in X then it would be pathological if there was any 1-dimensional vector subspace Y in which UY wasn't a neighborhood of the origin in at least some TVS topology on Y. But the only TVS topologies on Y are the Hausdorff Euclidean topology and the trivial topology, which is contained in the Euclidean topology. Consequently, it is natural to expect for UY to be a neighborhood of 0 in the Euclidean topology for all 1-dimensional vector subspaces Y, which is exactly the condition that U be absorbing in X. Thus, it is not surprising that all neighborhoods of the origin in all TVSs are necessarily absorbing. The reason why the Euclidean topology is distinguished is due to the defining requirement on TVS topologies that scalar multiplication be continuous when the scalar field is given the Euclidean topology.
    • This condition is equivalent to: For every xX, A ∩ span { x } is a neighborhood of 0 in span { x } = 𝕂 x when span { x } is given its unique Hausdorff TVS topology.
  7. A contains the origin and for every 1-dimensional vector subspace Y of X, AY is absorbing in the Y.
    • Here "absorbing" means absorbing according to any defining condition other than this one.
    • This shows that the property of being absorbing in X depends only on how A behaves with respect to 1 (or 0) dimensional vector subspaces of X.

If 𝕂 = ℝ then we can add to this list:

  1. The algebraic interior of A contains the origin (i.e. 0 ∈ iA).

If A is balanced then we can add to this list:

  1. For every xX, there exists a scalar c ≠ 0 such that xc A.[1]

If A is convex or balanced then we can add to this list:

  1. For every xX, there exists a positive real r > 0 such that r xA.
    • The proof that a balanced set A satisfying this condition is necessarily absorbing in X is almost immediate from the definition of a "balanced set".
    • The proof that a convex set A satisfying this condition is necessarily absorbing in X is less trivial (but not difficult). A detailed proof is given in this footnote[note 1] and a summary is given below.
      • Summary of proof: Observe that for any 0 ≠ yX, by assumption we may pick positive real r > 0 and R > 0 such that R yA and r (-y) ∈ A so that the convex set A ∩ ℝ y contains the open sub-interval (- r, R) y := { t y : - r < t < R, t ∈ ℝ}, which contains the origin (we also call A ∩ ℝ y an interval since any non-empty convex subset of is an interval). Give 𝕂 y its unique Hausdorff vector topology and we must show that A ∩ 𝕂 y is a neighborhood of the origin in 𝕂 y. If 𝕂 = ℝ then we're done, so assume that 𝕂 = ℂ. Note that the set S := (A ∩ ℝ y) ∪ (A ∩ ℝ (i y)) ⊆ A ∩ (ℂ y) is a union of two intervals, each of which contains an open sub-interval that contains the origin, and that the intersection of these two intervals is precisely the origin. So the convex hull of S, which is contained in the convex set A ∩ (ℂ y), clearly contains an open ball around the origin.
  2. For every xX, there exists a positive real r > 0 such that xr A.
    • This condition is equivalent to every xX belonging to the set (0, ∞) A := { r a : 0 < r < ∞, aA } = 0 < r < ∞ r A, which happens if and only if X = (0, ∞) A. One may also show that for any subset T of X, (0, ∞) T = X if and only if T ∩ (0, ∞) x ≠ ∅ for every xX.
  3. (0, ∞) A = X.
  4. For every xX, A ∩ (0, ∞) x ≠ ∅, where (0, ∞) x := { r x : 0 < r < ∞}.

Note: If 0 ∈ A (which is necessary for A to be absorbing) then it suffices to check any of the above conditions for all non-zero xX, rather than all xX.

Examples and sufficient conditions

For one set to absorb another

  • Let F : XY be a linear map between vector spaces and let BX and CY be balanced sets. Then C absorbs F(B) if and only if F−1(C) absorbs B.[2]

For a set to be absorbing

  • If X is a topological vector space (TVS) then any neighborhood of the origin in X is absorbing in X.
    • This fact is one of the primary motivations for even defining the property "absorbing in X."
  • In a semi normed vector space the unit ball is absorbing.
  • If D ≠ ∅ is a disk in X then span D =
    n=1
    nD
    so that in particular, D is an absorbing subset of span D.[3]
    • Thus if D is a disk in X, then X is absorbing in X if and only if span D = X.
  • The intersection of finite but nonempty family of absorbing sets is absorbing.
  • The union of a nonempty arbitrary family of absorbing sets is absorbing.
  • The image of an absorbing set under a surjective linear operator is again absorbing.
  • The inverse image of an absorbing set (in the codomain) under a linear operator is again absorbing (in the domain).

Properties

Every absorbing set contains the origin.

If D is an absorbing disk in a vector space X then there exists an absorbing disk E in X such that E + E D.[4]

See also

References

  1. Proof: Let X be a vector space over the field 𝕂, with 𝕂 being or , and endow the field 𝕂 with its usual normed Euclidean topology. Let A be a convex set such that for every zX, there exists a positive real r > 0 such that r zA. Since 0 ∈ A, if dim X = 0 then we're done so assume dim X ≠ 0. Clearly, every non-empty convex subset of the real line is an interval (possibly open, closed, or half-closed, and possibly bounded or unbounded, and possibly even degenerate (i.e. a single point)). Recall that the intersection of convex sets is convex so that for every 0 ≠ yX, the sets A ∩ 𝕂 y and A ∩ ℝ y are convex, where now the convexity of A ∩ ℝ y (which contains the origin and is contained in the line y) implies that A ∩ ℝ y is an interval contained in the line y := { r y : - ∞ < r < ∞}. Lemma: We will now prove that if 0 ≠ yX then the interval A ∩ ℝ y contains an open sub-interval that contains the origin. By assumption, since yX we can pick some R > 0 such that R yA and (since - yX) we can pick some r > 0 such that r (- y) ∈ A, where r (- y) = (- r) y and - r yR y (since y ≠ 0). Since A ∩ ℝ y is convex and contains the distinct points - r y and R y, it contains the convex hull of the points { - r y, R y}, which (in particular) contains the open sub-interval (- r, R) y := { t y : - r < t < R, t ∈ ℝ}, where this open sub-interval (- r, R) y contains the origin (since take t = 0, and note that - r < t = 0 < R), which proves the lemma. ∎ Now fix 0 ≠ xX, let Y := span { x } = 𝕂 x, and note that (since 0 ≠ xX was arbitrary), to prove that A is absorbing in X it is necessary and sufficient to show that AY is a neighborhood of the origin in Y when Y is given its usual Hausdorff Euclidean topology, where recall that this topology makes the map 𝕂 → 𝕂 x defined by cc x into a TVS-isomorphism. If 𝕂 = ℝ then the fact that the interval AY = A ∩ ℝ x contains an open sub-interval around the origin implies that AY is a neighborhood of the origin in Y = ℝ x so we're done. So assume that 𝕂 = ℂ. Note that i xY = ℂ x, where i = -1, and that Y = ℂ x = (ℝ x) + (ℝ (i x)) (so naively, x is the "x-axis" and ℝ (i x) is the "y-axis" of x so that the set (0, ∞) x (resp. (0, ∞) (i x)) is the strictly positive x-axis (resp. y-axis) while (-∞, 0) x (resp. (-∞, 0) (i x)) is the strictly negative x-axis (resp. y-axis)). Note that the set S := (A ∩ ℝ x) ∪ (A ∩ ℝ (i x)) is contained in the convex set AY, so that the convex hull of S is contained in AY. By the lemma, each of A ∩ ℝ x and A ∩ ℝ (i x) are line segments (i.e. intervals) with each segment containing the origin in an open sub-interval; moreover, they clearly intersect at the origin. Pick a real d > 0 such that (- d, d) x := { t x : - d < t < d, t ∈ ℝ } A ∩ ℝ x and (- d, d) ix := { t ix : - d < t < d, t ∈ ℝ } A ∩ ℝ (i x). Let N denote the convex hull of [(- d, d) x] ∪ [(- d, d) ix], which is contained in the convex hull of S and thus also contained in the convex set AY. So to finish the proof, it suffices to show that N is a neighborhood of 0 in Y. It is easily verified that N contains the open ball Bd/2 x := { c x : c ∈ 𝕂 and |c| < d/2} of radius d/2 centered at the origin of Y = ℂ x. This shows that AY is a neighborhood of the origin in Y = ℂ x, as desired. ∎
  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 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. 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. 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. 16. Providence, R.I.: American Mathematical Society. ISBN 9781470424831. OCLC 185095773.
  • Dineen, Seán (1981). Complex Analysis in Locally Convex Spaces. North-Holland Mathematics Studies. 57. Amsterdam New York New York: North-Holland Pub. Co., Elsevier Science Pub. Co. ISBN 978-0-08-087168-4. OCLC 16549589.CS1 maint: ref=harv (link)
  • Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. 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.
  • Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. OCLC 316549583.
  • Hogbe-Nlend, Henri; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. 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. 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. 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. 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 (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 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. 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.
  • Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
  • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 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. 53. Cambridge University Press. p. 4.
  • 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.
  • 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. 3. New York: Springer-Verlag. p. 11. ISBN 0-387-98726-6.
  • 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.
  • 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.
  • Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.
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