Non-Associative Local Lie Groups
... [6], who showed that every local Lie group contains a neighborhood of the identity which is homeomorphic to a neighborhood of the identity of a global Lie group; see also [28; Theorem 84]. Cartan’s result provides a global version Lie’s Third Fundamental Theorem — every Lie algebra is the Lie algebr ...
... [6], who showed that every local Lie group contains a neighborhood of the identity which is homeomorphic to a neighborhood of the identity of a global Lie group; see also [28; Theorem 84]. Cartan’s result provides a global version Lie’s Third Fundamental Theorem — every Lie algebra is the Lie algebr ...
Affine Decomposition of Isometries in Nilpotent Lie Groups
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
... Let’s move some steps towards mathematics. The result we will prove is stated in its fully formal form as follows Theorem 1.1. Let (N1 , d1 ) and (N2 , d2 ) be two connected nilpotent metric Lie groups. Then any isometry F : N1 → N2 is affine. In the preliminaries section we are going to go through ...
Minimal Totally Disconnected Spaces
... (b) If X is D-dosed,thenit hasa denseset of isolatedpoints. Proof. (a). If every suchspacehas at least one isolatedpoint, then the set I of isolatedpointsof X must be infinite, for, otherwise,X • I would be a clopenset satisfyingzero-dimensional (i) and wouldhavean isolated point p, but thenp wouldb ...
... (b) If X is D-dosed,thenit hasa denseset of isolatedpoints. Proof. (a). If every suchspacehas at least one isolatedpoint, then the set I of isolatedpointsof X must be infinite, for, otherwise,X • I would be a clopenset satisfyingzero-dimensional (i) and wouldhavean isolated point p, but thenp wouldb ...
Functional Analysis Lecture Notes
... 1.10 Theorem. Every basis of a given linear space has the same cardinality. Proof. Let S and T be bases for a linear space X over K. We shall demonstrate that there exists an injection Φ : S → T . This will be enough, since the roles of S and T can be interchanged to produce an injection on T into S ...
... 1.10 Theorem. Every basis of a given linear space has the same cardinality. Proof. Let S and T be bases for a linear space X over K. We shall demonstrate that there exists an injection Φ : S → T . This will be enough, since the roles of S and T can be interchanged to produce an injection on T into S ...
General Topology - Fakultät für Mathematik
... lower bound of a family of topologies on X. In fact, if (Oi )i∈I is a family of toplogies on X then 1.3.2, applied to id: T (X, Oi ) → X shows that the finest topology coarser than all Oi is given by O = i∈I Oi . An important special case of final topologies is the quotient topology: 1.3.5 Definitio ...
... lower bound of a family of topologies on X. In fact, if (Oi )i∈I is a family of toplogies on X then 1.3.2, applied to id: T (X, Oi ) → X shows that the finest topology coarser than all Oi is given by O = i∈I Oi . An important special case of final topologies is the quotient topology: 1.3.5 Definitio ...