Absolutely convex set

A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.

Definition

The light gray area is the absolutely convex hull of the cross.

If $S$ is a subset of a real or complex vector space $X$ , then we call $S$ a disk and say that $S$ is disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied:

$S$ is convex and balanced;
for any scalars $a$ and $b$ satisfying $| a | + | b | \leq 1$ , $aS + bS \subseteq S$ ;
for all scalars $a$ , $b$ , and $c$ satisfying |a| + |b| ≤ |c|, $aS + bS \subseteq cS$ ;
for any scalars $a 1, ..., a n$ satisfying $\sum _{i=1}^{n}|a_{i}|\leq 1$ , $a_{1}S+\cdots +a_{n}S\subseteq S$ ;
for any scalars $c$ , $a 1, ..., a n$ satisfying $\sum _{i=1}^{n}|a_{i}|\leq |c|$ , $a_{1}S+\cdots +a_{n}S\subseteq cS$ ;

Recall that the smallest convex (resp. balanced) subset of $X$ containing a set is called the convex hull (resp. balanced hull) of that set and is denoted by $co(S)$ (resp. $bal(S)$ ).

Similarly, we define the disked hull, the absolute convex hull, or the convex balanced hull of a set $S$ is defined to be the smallest disk (with respect to subset inclusion) containing $S$ .[1] The disked hull of $S$ will be denoted by $disk S$ or $cobal S$ and it is equal to each of the following sets:

co(bal(S)), which is the convex hull of the balanced hull of S; thus, cobal(S) = co(bal(S));
- Note however that in general, $cobal(S) \neq bal(co(S))$ , even in finite dimensions.
the intersection of all disks containing $S$
$\left\{\sum _{i=1}^{n}\lambda _{i}x_{i}:n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}|\lambda _{i}|\leq 1\right\},$ where the $λ i$ are elements of the underlying field.

Sufficient conditions

The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.
if $D$ is a disk in $X$ , then $X$ is absorbing in $X$ if and only if $span D = X$ .[2]

Properties

If $S$ is an absorbing disk in a vector space $X$ then there exists an absorbing disk $E$ in $X$ such that $E + E \subseteq S$ .[3]
The convex balanced hull of $S$ contains both the convex hull of $S$ and the balanced hull of $S$ .
The absolutely convex hull of a bounded set in a topological vector space is again bounded.
If D is a bounded disk in a TVS X and if x_• = (x_i)^∞
_i=1 is a sequence in D, then the partial sums s_• = (s_n)^∞
_i=1 are Cauchy, where for all n, s_n := ∑ⁿ
_i=1 2^-i x_i .[4]
- In particular, if in addition $D$ is a sequentially complete subset of $X$ , then this series $s •$ converges in $X$ to some point of $D$ .

Examples

Although $cobal(S) = co(bal(S))$ , the convex balanced hull of $S$ is not necessarily equal to the balanced hull of the convex hull of $S$ .[1] For an example where $cobal(S) \neq bal(co(S))$ , let $X$ be the real vector space $ℝ 2$ and let $S := { (-1, 1), (1, 1)$ }. Then $bal(co(S))$ is a strict subset of cobal(S) that is not even convex. In particular, this example also shows that the balanced hull of a convex set is not necessarily convex. To see this, note that $cobal(S)$ is equal to the closed square in $X$ with vertices $(-1, 1), (1, 1), (-1, -1)$ , and $(-1, 1)$ while $bal(co(S))$ is a closed "hour glass shaped" shaped subset that intersects the $x$ -axis at the origin and is the union of two triangles: one whose vertices are the origin along with $S$ and the other triangle whose vertices are the origin along with $-S = { (-1, -1), (1, -1)$ }.

References

Trèves 2006, p. 68.
Narici 2011, pp. 67-113.
Narici 2011, pp. 149-153.
Narici 2011, p. 471.

Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 4–6.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, H.H. (1999). Topological vector spaces. Springer-Verlag Press. p. 39.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[FOOTNOTETrèves200668-1] Trèves 2006, p. 68.

[FOOTNOTENarici201167-113-2] Narici 2011, pp. 67-113.

[FOOTNOTENarici2011149-153-3] Narici 2011, pp. 149-153.

[FOOTNOTENarici2011471-4] Narici 2011, p. 471.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-Valued Hahn-Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property