Lucas Borboleta - Tag - espaceAu fil du temps, l'auteur entreprend de partager quelques travaux d'amateur en mathématiques, physique, photographie, etc.
Over time, the author undertakes to share some amateur work in mathematics, physics, photography, etc.2024-03-21T16:26:49+01:00Lucas Borboletaurn:md5:0c9ce0ab0af019c8bd0152e323e3a939DotclearInner Product Space Characterization by Isometry Group over Unity Sphereurn:md5:759e4c08c9a6162ddd26d565400fc6ba2013-12-01T19:08:00+01:00LucasMathématiquesespaceeuclidiengroupeorthogonalité <p>Why “power 2” in the Pythagorean theorem ? </p>
<p>Let us assume a geometry over the set E considered as 1) a vector space over the real field ; 2) with finite dimensions ; 3) equiped with a norm. </p>
<p>Which additional axioms have to be added in order to conclude that E is euclidean, or equivalently, that its norm is deriving from an inner product. Such additionnal axioms are searched in topology and in group formulations, as opposed to calculational ones, like the “polarisation identity”.</p>
<p>The attached document gathers theorems that proof the following axiom solution: 4) if a group G of isometries allows transportation between any pair of points of the unity sphere of E, then E is euclidean.</p>
<p><a href="http://lucas.borboleta.blog.free.fr/public/Inner-Product-Space-Characterization-by-Isometric-Group-over-Unity-Sphere/Borboleta_2013_Inner-Product-Space-Characterization-by-Isometric-Group-over-Unity-Sphere.pdf">Borboleta_2013_Inner-Product-Space-Characterization-by-Isometric-Group-over-Unity-Sphere.pdf</a></p>
<p><img src="http://lucas.borboleta.blog.free.fr/public/Inner-Product-Space-Characterization-by-Isometric-Group-over-Unity-Sphere/.500px-Pythagorean_theorem_abc.svg_s.jpg" alt="" title="500px-Pythagorean_theorem_abc.svg.png, déc. 2013" /></p>http://lucas.borboleta.blog.free.fr/index.php?post/2013/12/01/Inner-Product-Space-Characterization-by-Isometry-Group-over-Unity-Sphere2#comment-formhttp://lucas.borboleta.blog.free.fr/index.php?feed/atom/comments/4780648Return on the 1932 proof of the Auerbach Theorem about Bounded Linear Groupsurn:md5:ff6227d38ff0d73e83846c125bab58822013-11-24T17:50:00+01:00LucasMathématiquesespacegroupeinvariantnormequadratique In 1932, Auerbach proved that, in any finite vector normed space, each bounded linear group <span style="background- line-height: 20.15625px;">left invariant a quadratic and positive form. This work revisits his proof with modern </span><span style="background- line-height: 20.15625px;">notation, and aims at paying attention to weaknesses: </span><a href="http://lucas.borboleta.blog.free.fr/public/Borboleta_2013_Return-on-Auerbach-Theorem-for-bounded-linear-group/Borboleta_2013_Return-on-Auerbach-Theorem-for-bounded-linear-group.pdf" style="background- line-height: 20.15625px;">Borboleta_2013_Return-on-Auerbach-Theorem-for-bounded-linear-group.pdf</a><span style="background- line-height: 20.15625px;">.</span>http://lucas.borboleta.blog.free.fr/index.php?post/2013/11/24/Auerbach#comment-formhttp://lucas.borboleta.blog.free.fr/index.php?feed/atom/comments/4779754