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