Absorbing and balanced sets

January 28th, 2006 ~ Posted in: Mathematics

If X is a vector space over a field \mathbb{F} and A \subseteq X, we have the following definitions to consider:

  1. A is convex if t y + (1-t)z \in A whenever y,z \in A and 0 < t < 1. Here \mathbb{F} is \R or \C.
  2. If \alpha A \subseteq A whenever |\alpha| \leq 1, then A is balanced.
  3. The set A is absorbing if, for each x \in X, there is a positive number s_x such that x \in tA whenever t > s_x.

An introduction to Banach Space Theory. Robert E. Megginson

Some consequences of sets’ having these properties are well known or intuitively obvious, especially concerning convexity. Here are some interesting questions, from the same source:

  1. Identify all the balanced subsets of \C. Do the same for \R^2.
  2. Suppose that A is balanced. Prove that A is absorbing iff for each x \in X there is a positive number t_x such that x \in t_x A.
  3. Prove that each convex absorbing subset of \C contains a neighborhood of 0. Is this true for nonconvex absorbing subsets of \C?

In relation to the last question above, consider the polar graph of the relation 2\pi r - (2\pi - \theta) \cos \theta = 0

Mathematics ~

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