balanced setBalancedSet2005-10-28 15:29:152005-10-29 00:44:15Definitionbalanced subsetbalanced hullbalanced evelopecircled\'equilibr\'e% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition} \cite{rudin_fap,edwards, horvath, cristescu}
Let $V$ be a vector space over $\sR$ (or $\sC$),
and let $S$ be a subset of $V$. If $\lambda S\subset S$ for all scalars $\lambda$ such
that $|\lambda|\le 1$, then $S$ is a {\bf balanced set} in $V$.
The {\bf balanced hull} of $S$,
denoted by $\operatorname{eq}(S)$, is the smallest
balanced set containing $S$.
In the above,
$\lambda S = \{ \lambda s\mid s\in S\}$,
and $|\cdot|$ is the absolute value (in $\sR$),
or the modulus of a complex number (in $\sC$).
\subsubsection{Examples and properties}
\begin{enumerate}
\item Let $V$ be a normed space with norm $||\cdot||$. Then the unit ball
$\{v\in V\mid ||v||\le 1\}$ is a balanced set.
\item Any vector subspace is a balanced set. Thus, in $\sR^3$, lines and planes passing
through the origin are balanced sets.
\end{enumerate}
\subsubsection{Notes}
A balanced set is also sometimes called {\bf circled} \cite{horvath}.
The term {\bf balanced evelope} is also used for the balanced hull \cite{edwards}.
Bourbaki uses the term {\bf \'equilibr\'e} \cite{edwards}, c.f. $\operatorname{eq}(A)$
above. In \cite{reed}, a balanced set is defined as above, but with the condition $|\lambda|=1$ instead of $|\lambda|\le 1$.
\begin{thebibliography}{9}
\bibitem{rudin_fap}
W. Rudin, \emph{Functional Analysis},
McGraw-Hill Book Company, 1973.
\bibitem{edwards} R.E. Edwards, \emph{Functional Analysis: Theory and Applications},
Dover Publications, 1995.
\bibitem{horvath} J. Horv\'ath, \emph{Topological Vector Spaces and Distributions},
Addison-Wsley Publishing Company, 1966.
\bibitem{cristescu} R. Cristescu, \emph{Topological vector spaces},
Noordhoff International Publishing, 1977.
\bibitem{reed} M. Reed, B. Simon,
\emph{Methods of Modern Mathematical Physics: Functional Analysis I},
Revised and enlarged edition, Academic Press, 1980.
\end{thebibliography}