is the union of all balanced subsets of if while if Examples The empty set is a balanced set. As is any vector subspace of any (real or complex) vector space. In particular, is always a balanced set. Any non-empty set that does not contain the origin is not balanced and furthermore, the balanced core of such a set will equal the empty set. Normed and topological vectors spaces The open and closed balls centered at the origin in a normed vector space are balanced sets. If is a seminorm (or norm) on a vector space then for any constant the set is balanced. If is any subset and then is a balanced set. In particular, if is any balanced neighborhood of the origin in a topological vector space then Balanced sets in and Let be the field real numbers or complex numbers let denote the absolute value on and let denotes the vector space over So for example, if is the field of complex numbers then is a 1-dimensional complex vector space whereas if then is a 1-dimensional real vector space. The balanced subsets of are exactly the following: for some real for some real Consequently, both the balanced core and the balanced hull of every set of scalars is equal to one of the sets listed above. The balanced sets are itself, the empty set and the open and closed discs centered at zero. Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do. As a result, and are entirely different as far as scalar multiplication is concerned. Balanced sets in Throughout, let (so is a vector space over ) and let is the closed unit ball in centered at the origin. If is non-zero, and then the set is a closed, symmetric, and balanced neighborhood of the origin in More generally, if is closed subset of such that then is a closed, symmetric, and balanced neighborhood of the origin in This example can be generalized to for any integer Let be the union of the line segment between the points and and the line segment between and Then is balanced but not convex or absorbing. However, For every let be any positive real number and let be the (open or closed) line segment in between the points and Then the set is a balanced