## You are here

Homeboundedness in a topological vector space generalizes boundedness in a normed space

## Primary tabs

# boundedness in a topological vector space generalizes boundedness in a normed space

Boundedness in a topological vector space is a generalization of boundedness in a normed space.

Suppose $(V,\|\cdot\|)$ is a normed vector space over $\mathbbmss{C}$, and suppose $B$ is bounded in the sense of the parent entry. Then for the unit ball

$B_{1}(0)=\{v\in V:\|v\|<1\}$ |

there exists some $\lambda\in\mathbbmss{C}$ such that $B\subseteq\lambda B_{1}(0)$. Using this result, it follows that

$B\subseteq B_{{|\lambda|}}(0).$ |

Major Section:

Reference

Type of Math Object:

Result

## Mathematics Subject Classification

46-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Corrections

proof does not match title by Mathprof ✓

Still wrong by CWoo ✘

Still wrong by CWoo ✘

Error still present by CWoo ✓

Still wrong by CWoo ✘

Still wrong by CWoo ✘

Error still present by CWoo ✓