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\begin{document}
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\title{The Hahn-Banach Theorem \\ for Real Vector Spaces}
\author{Gertrud Bauer \\ \url{http://www.in.tum.de/~bauerg/}}
\maketitle
\begin{abstract}
The Hahn-Banach Theorem is one of the most fundamental results in functional
analysis. We present a fully formal proof of two versions of the theorem,
one for general linear spaces and another for normed spaces. This
development is based on simply-typed classical set-theory, as provided by
Isabelle/HOL.
\end{abstract}
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\section{Preface}
This is a fully formal proof of the Hahn-Banach Theorem. It closely follows
the informal presentation given in Heuser's textbook \cite[{\S} 36]{Heuser:1986}.
Another formal proof of the same theorem has been done in Mizar
\cite{Nowak:1993}. A general overview of the relevance and history of the
Hahn-Banach Theorem is given by Narici and Beckenstein \cite{Narici:1996}.
\medskip The document is structured as follows. The first part contains
definitions of basic notions of linear algebra: vector spaces, subspaces,
normed spaces, continuous linear-forms, norm of functions and an order on
functions by domain extension. The second part contains some lemmas about the
supremum (w.r.t.\ the function order) and extension of non-maximal functions.
With these preliminaries, the main proof of the theorem (in its two versions)
is conducted in the third part. The dependencies of individual theories are
as follows.
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\part {Basic Notions}
\input{Bounds}
\input{VectorSpace}
\input{Subspace}
\input{NormedSpace}
\input{Linearform}
\input{FunctionOrder}
\input{FunctionNorm}
\input{ZornLemma}
\clearpage
\part {Lemmas for the Proof}
\input{HahnBanachSupLemmas}
\input{HahnBanachExtLemmas}
\input{HahnBanachLemmas}
\clearpage
\part {The Main Proof}
\input{HahnBanach}
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\end{document}