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2 \documentclass[10pt,a4paper,twoside]{article} |
2 \documentclass[10pt,a4paper,twoside]{article} |
3 |
3 \usepackage{graphicx} |
4 \usepackage{latexsym,theorem} |
4 \usepackage{latexsym,theorem} |
5 \usepackage{isabelle,isabellesym} |
5 \usepackage{isabelle,isabellesym} |
6 \usepackage{pdfsetup} %last one! |
6 \usepackage{pdfsetup} %last one! |
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8 \isabellestyle{it} |
8 \isabellestyle{it} |
15 \begin{document} |
15 \begin{document} |
16 |
16 |
17 \pagestyle{headings} |
17 \pagestyle{headings} |
18 \pagenumbering{arabic} |
18 \pagenumbering{arabic} |
19 |
19 |
20 \title{The Hahn-Banach Theorem for Real Vector Spaces} |
20 \title{The Hahn-Banach Theorem \\ for Real Vector Spaces} |
21 \author{Gertrud Bauer \\ \url{http://www.in.tum.de/~bauerg/}} |
21 \author{Gertrud Bauer \\ \url{http://www.in.tum.de/~bauerg/}} |
22 \maketitle |
22 \maketitle |
23 |
23 |
24 \begin{abstract} |
24 \begin{abstract} |
25 The Hahn-Banach Theorem is one of the most fundamental results in functional |
25 The Hahn-Banach Theorem is one of the most fundamental results in functional |
42 \cite{Nowak:1993}. A general overview of the relevance and history of the |
42 \cite{Nowak:1993}. A general overview of the relevance and history of the |
43 Hahn-Banach Theorem is given in \cite{Narici:1996}. |
43 Hahn-Banach Theorem is given in \cite{Narici:1996}. |
44 |
44 |
45 \medskip The document is structured as follows. The first part contains |
45 \medskip The document is structured as follows. The first part contains |
46 definitions of basic notions of linear algebra: vector spaces, subspaces, |
46 definitions of basic notions of linear algebra: vector spaces, subspaces, |
47 normed spaces, continuous linearforms, norm of functions and an order on |
47 normed spaces, continuous linear-forms, norm of functions and an order on |
48 functions by domain extension. The second part contains some lemmas about the |
48 functions by domain extension. The second part contains some lemmas about the |
49 supremum (w.r.t.\ the function order) and extension of non-maximal functions. |
49 supremum (w.r.t.\ the function order) and extension of non-maximal functions. |
50 With these preliminaries, the main proof of the theorem (in its two versions) |
50 With these preliminaries, the main proof of the theorem (in its two versions) |
51 is conducted in the third part. |
51 is conducted in the third part. The dependencies of individual theories are |
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52 as follows. |
52 |
53 |
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54 \begin{center} |
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55 \includegraphics[scale=0.7]{session_graph} |
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56 \end{center} |
53 |
57 |
54 \clearpage |
58 \clearpage |
55 \part {Basic Notions} |
59 \part {Basic Notions} |
56 |
60 |
57 \input{Bounds} |
61 \input{Bounds} |