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 ${\{{z}_{i}\}}_{i=1}^{\mathrm{\infty}}$ is a sequence in $B$, and ${\{{\lambda}_{i}\}}_{i=1}^{\mathrm{\infty}}$ is a sequence of scalars (in $\mathbb{R}$ or $\u2102$), such that ${\lambda}_{i}\to 0$, then ${\lambda}_{i}{z}_{i}\to 0$ in $V$.
References
 1 W. Rudin, Functional Analysis^{}, McGrawHill Book Company, 1973.
 2 R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
Title  sequential characterization of boundedness 

Canonical name  SequentialCharacterizationOfBoundedness 
Date of creation  20130322 13:48:17 
Last modified on  20130322 13:48:17 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  8 
Author  bwebste (988) 
Entry type  Theorem 
Classification  msc 4600 