sequential characterization of boundedness

Theorem [1, 2] A set $B$ in a real (or possibly complex) topological vector space $V$ is bounded (http://planetmath.org/BoundedSetInATopologicalVectorSpace) if and only if the following condition holds:

If $\{z_{i}\}_{i=1}^{\infty}$ is a sequence in $B$ , and $\{\lambda_{i}\}_{i=1}^{\infty}$ is a sequence of scalars (in $\mathbb{R}$ or $\mathbb{C}$ ), such that $\lambda_{i}\to 0$ , then $\lambda_{i}z_{i}\to 0$ in $V$ .

References

1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
2 R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.

Title	sequential characterization of boundedness
Canonical name	SequentialCharacterizationOfBoundedness
Date of creation	2013-03-22 13:48:17
Last modified on	2013-03-22 13:48:17
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	8
Author	bwebste (988)
Entry type	Theorem
Classification	msc 46-00