Absolutely convex set

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:

1. S is convex and balanced;
2. for any scalars a and b satisfying |a| + |b| ≤ 1, aS + bS ⊆ S;
3. for all scalars a, b, and c satisfying |a| + |b| ≤ |c|, aS + bS ⊆ cS;
4. for any scalars a₁, ..., a_n satisfying ${\displaystyle \sum _{i=1}^{n}|a_{i}|\leq 1}$, ${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq S}$;
5. for any scalars c, a₁, ..., a_n satisfying ${\displaystyle \sum _{i=1}^{n}|a_{i}|\leq |c|}$, ${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq cS}$;

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:

1. 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) ≠ bal(co(S)), even in finite dimensions,
2. the intersection of all disks containing S,
3. ${\displaystyle \left\{\sum _{i=1}^{n}\lambda _{i}x_{i}:n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}|\lambda _{i}|\leq 1\right\},}$ where the λ_i are elements of the underlying field.

• 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]

• If S is an absorbing disk in a vector space X then there exists an absorbing disk E in X such that E + E ⊆ 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.

The Wikibook Algebra has a page on the topic of: Vector spaces

• Absorbing set
• Balanced set
• Bounded set (topological vector space)
• Convex set
• Star domain
• Symmetric set
• Vector (geometric), for vectors in physics
• Vector field

• 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.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.