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:

1. 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 SequentialCharacterizationOfBoundedness 2013-03-22 13:48:17 2013-03-22 13:48:17 bwebste (988) bwebste (988) 8 bwebste (988) Theorem msc 46-00