Si la constante est incluse dans X, l'hyperplan inclut l'origine et est un sous-espace sinon, il s'agit d'un ensemble affine coupant l'axe Y au point (0, ). Dans l'espace d'entre-sortie (p + 1), (X, Y) reprsente un hyperplan. (2) If 0 f V, then ker f is always a hyperplane ( a maximal proper subspace of V ) (3) If dim F V n <, then with the same notation and assumptions as above dim ker f n 1. La dimension de ce tore est le rang de lorbite G y. Comprendre la projection linaire dans Les lments de l'apprentissage statistique. (1) If V F is any vector space and 0 f V is any (non-zero) linear functional, then f is always onto. Si on est dans un espace de dimension n, cela signifie que lhyperplan est de dimension n-1. Il existe un voisinage ouvert de y dans X, isomorphe au produit direct dun espace affine par un tore. Hyperplan tangent à une hypersurface algébrique de degré quelconque en un point régulier. Les 4 varits T, T/W, G et Y permettent de dfinir un diagramme commutatif de la faon suivante : o est la premire projection de Y sur G. TOPOLOGICAL PRELIMINARIES Topological definitions not included here can be found in. En algèbre linéaire, un hyperplan affineest un sous-espace affinede codimension1. Hyperplan tangent à une conique en un point régulier. Some of these results have been presented without proof in the Choquet Seminar. In Section 3 examples are given of compact convex sets such that their sets of extreme points satisfy certain topological properties. The proof depends upon the results of Corson and a metrization theorem of Jayne. It will be shown here that the set of extreme points of a compact convex set X is a Baire subset (or more generally a Z-Souslin subset) of X if and only if X is metrizable. In the importance (to integral representation theory) of the Baire subsets rather than the K-Borelian subsets of a compact convex set is emphasized. By applying an affine transformation if necessary, wc can assUIll( thflt.

Examples are given of compact convex sets such that their sets of extreme points satisfy certain topological properties. Considrons le problme suivant: trouver lhyperplan approximant. invariants est un hyperplan donc de soit et est un automorphisme de P. It is shown that the set of extreme points of X is a Bain subset (or more generally a Z-Souslin subset) of X if and only if X is metrizable. La projection orthogonale par rapport à F, ’est la projetion sur F parallèlement à. affine manifol d an d M i s an n-dimensiona l compac t algebrai c manifol d wit h. Dixmier Received NovemLet X be a compact convex subset of a HausdorfI locally convex real topological vector space. given an d le t ther e b e q > n hyperplane s D 3 C Pn i n genera l.

Bernard (Tour 46), Paris 5, France Communicated by J. While the interior corresponds to the inequalities becoming strict (increasing sequences).JOURNAL OF FUNCTIONAL ANALYSIS 11, 385-392 (1972) A Criterion for the Metrizability of a Compact Convex Set in Terms of the Set of Extreme Points* BRENDA MACGIBBON+ Universite’ Paris VI, Matkimatiques, 11, Quai St.

Projection sur un hyperplan affine