Topological Rings and Infinite Matrices |
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Page 9
... integers , with the following propert- ies : 1 . n ( a + p ) = na + nß . 2 . ( n + m ) α = na + ma 3 . n ( ma ) = ( nm ) % . 4. 1.8 = 0 . If we assign to the integers the discrete topology , that is , each element is its own ...
... integers , with the following propert- ies : 1 . n ( a + p ) = na + nß . 2 . ( n + m ) α = na + ma 3 . n ( ma ) = ( nm ) % . 4. 1.8 = 0 . If we assign to the integers the discrete topology , that is , each element is its own ...
Page 31
... integers . Into this group a topology may be introduced in a simple way , as , indeed , may be done in any quotient group where the subgroup is closed . The elements of T may be thought of as sets R + x , where R is the set of integers ...
... integers . Into this group a topology may be introduced in a simple way , as , indeed , may be done in any quotient group where the subgroup is closed . The elements of T may be thought of as sets R + x , where R is the set of integers ...
Page 50
... integers , the matrix ( ; ,, S ; § ; ,, ♪ ̧ ̧ ‚ ...... .. ) is non - singular . Proof : Obviously the columns of the matrix form a basis for the whole space . Lemma 4.11 : Every matrix of the form A = ( S ,, S ,, ... , § ; . , ‚ § , S ...
... integers , the matrix ( ; ,, S ; § ; ,, ♪ ̧ ̧ ‚ ...... .. ) is non - singular . Proof : Obviously the columns of the matrix form a basis for the whole space . Lemma 4.11 : Every matrix of the form A = ( S ,, S ,, ... , § ; . , ‚ § , S ...
Common terms and phrases
abelian topological group additive group axiom of countability bourhood columns compact and satisfy compact commutative group continuous function correspondence is continuous correspondence is preserved countability axiom countable set d₂ defined definition 2.9 denote exists a neighbourhood finitely non-zero rows form a basis G into G G₂ give all possible group with operator Hausdorff space homomorphism of G infinite basis infinite direct sum infinite matrices infinitely non-zero integers inuous lemma Let G linear subspace linear transformations matrices of M(V matrix theory neigh neighbourhood of zero neighbourhoods in G non-singular matrices open set operator homomorph operator ring possible orders preserved under addition principal diagonal principal ideal ring Proof real numbers ring of homomorphisms ring of infinite ring of matrices ring of operators Rings and Infinite satisfying the second second axiom sequence subring subset Suppose lim theorem 4.1 theory of infinite thesis topological ring U₂ University of Wisconsin vector space whence