Topological Rings and Infinite Matrices |
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Page 16
... Definition 2.9 : Let G ,, G2 be closed R - admissable subgroups of an abel- ian topological group G with operator ring R such that 1. G , G2 = [ 0 ] . 2 . 3 . ^ 2 G , + G2 = G. + If U , is an open set in G ,, U , open in G2 , then U ...
... Definition 2.9 : Let G ,, G2 be closed R - admissable subgroups of an abel- ian topological group G with operator ring R such that 1. G , G2 = [ 0 ] . 2 . 3 . ^ 2 G , + G2 = G. + If U , is an open set in G ,, U , open in G2 , then U ...
Page 18
... meaning regardless of whether the situation involved is that of definition 2.9 , or definition 2.9 . We have yet to show that in the case of locally compact groups satisfying the second countability axiom definition 2.9 reduces to that ...
... meaning regardless of whether the situation involved is that of definition 2.9 , or definition 2.9 . We have yet to show that in the case of locally compact groups satisfying the second countability axiom definition 2.9 reduces to that ...
Page 19
... definition 2.9 , then GG , G2 • Now GG ,, GG2 , and G * , G * satisfy ( 3 ) of definition 2.9 . But , using the isomorphism , G ,, G2 satisfy ( 3 ) of definition 2.9 . We now establish a definition for countably infinite direct sums .
... definition 2.9 , then GG , G2 • Now GG ,, GG2 , and G * , G * satisfy ( 3 ) of definition 2.9 . But , using the isomorphism , G ,, G2 satisfy ( 3 ) of definition 2.9 . We now establish a definition for countably infinite direct sums .
Common terms and phrases
abelian topological group addition and multiplication 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 infinite basis infinite direct sum infinite matrices infinitely non-zero integers inuous lemma Let G linear subspace linear transformations matrix theory neigh neighbourhood of zero neighbourhoods in G non-singular matrices noted 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 Suppose lim theorem 4.1 theory of infinite thesis topological ring U₂ University of Wisconsin vector space whence