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\begin{document}

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\section*{bornological topological vector space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{maps_on_bornological_spaces}{Maps on bornological spaces}\dotfill \pageref*{maps_on_bornological_spaces} \linebreak
\noindent\hyperlink{relation_to_banach_spaces}{Relation to Banach spaces}\dotfill \pageref*{relation_to_banach_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Linear operators on [[normed space|normed spaces]] are continuous precisely iff they are bounded. A bornological space retains this property by definition.

The discussion below is about bornological CVSes, but there is a more general notion of [[bornological space]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A [[locally convex]] [[topological vector space]] $E$ is \textbf{bornological} if every circled, convex subset $A \subset E$ that absorbs every [[bounded set]] in $E$ is a neighbourhood of $0$ in $E$. Equivalently every [[seminorm]] that is bounded on bounded sets is continuous.

The \textbf{bornology} of a given [[TVS]] is the family of bounded subsets.

Given a locally convex [[TVS]] $E$ with initial topology $T_0$, there is a finest topology $T$ such that the family of bounded subsets of $T$ coincides with $T_0$. The space $E$ equipped with the topology $T$ is called the \textbf{bornologification} of $E$, or the \textbf{bornological space associated with} $(E, T_0)$

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{maps_on_bornological_spaces}{}\subsubsection*{{Maps on bornological spaces}}\label{maps_on_bornological_spaces}

\begin{theorem}
\label{}\hypertarget{}{}
Let $U$ be a linear map from a bornological space $E$ to any locally convex [[TVS]], then the following statements are equivalent:

\begin{itemize}%
\item $U$ is continuous,


\item $U$ is bounded on bounded sets,


\item $U$ maps [[null sequence|null sequences]] to null sequences.



\end{itemize}
\end{theorem}
\hypertarget{relation_to_banach_spaces}{}\subsubsection*{{Relation to Banach spaces}}\label{relation_to_banach_spaces}

Every [[inductive limit]] of [[Banach spaces]] is a bornological vector space. (\hyperlink{AlpaySalomon13}{Alpay-Salomon 13, prop. 2.3})

Conversely, every bornological vector space is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete (\hyperlink{SchaeferWolff99}{Schaefer-Wolff 99})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Every metrizible locally convex space is bornological, that is every [[Fréchet space]] and thus every [[Banach space]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[bornological set]]


\item [[bornological group]]


\item [[bornological space]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia about \href{http://en.wikipedia.org/wiki/Bornological_space}{bornological spaces}


\item H. H. Schaefer with M. P. Wolff, section 8 of \emph{Topological vector spaces}, Springer 1999


\item Daniel Alpay, Guy Salomon, \emph{On algebras which are inductive limits of Banach spaces} (\href{http://arxiv.org/abs/1302.3372}{arXiv:1302.3372})



\end{itemize}
Discussion of bornological vector spaces forming a [[quasi-abelian category]] is in

\begin{itemize}%
\item [[Fabienne Prosmans]], [[Jean-Pierre Schneiders]], \emph{A homological study of bornological spaces}, December 2000, Prepublications Mathematiques de l'Universite Paris 13, 46 (\href{http://www.analg.ulg.ac.be/jps/rec/hsbs.pdf}{pdf})

\end{itemize}
with review and generalization to [[bornological abelian groups]] in

\begin{itemize}%
\item [[Federico Bambozzi]], section 1 of \emph{On a generalization of affinoid varieties} (\href{http://arxiv.org/abs/1401.5702}{arXiv:1401.5702})

\end{itemize}
[[!redirects bornologification]]

[[!redirects bornological vector space]] [[!redirects bornological vector spaces]] [[!redirects bornological topological vector spaces]] [[!redirects Bornological topological vector space]]



\end{document}