complete notes
... Let �� be the biggest number in the set {�0 � � � � � �� } satisfying �� < ��+1 , and let �� be the smallest number in {�0 � � � � � �� } satisfying ��+1 < �� . Since �� < �� we have V �� ⊆ V�� . By Lemma 10.3 there exists an open set V such that V �� ⊆ V and V ⊆ V�� . We set V��+1 := V . One can as ...
... Let �� be the biggest number in the set {�0 � � � � � �� } satisfying �� < ��+1 , and let �� be the smallest number in {�0 � � � � � �� } satisfying ��+1 < �� . Since �� < �� we have V �� ⊆ V�� . By Lemma 10.3 there exists an open set V such that V �� ⊆ V and V ⊆ V�� . We set V��+1 := V . One can as ...
Section 12.2. The Tychonoff Product Theorem
... Definition. Let {(Xλ , Tλ )}λ∈Λ be a collection of topological spaces indexed by Q arbitrary set Λ. The product topology on the Cartesian product λ∈Λ Xλ is the Q topology that has as a base sets of the form λ∈Λ Oλ where each Oλ ∈ Tλ is open and Oλ = Xλ except for finitely many λ. ...
... Definition. Let {(Xλ , Tλ )}λ∈Λ be a collection of topological spaces indexed by Q arbitrary set Λ. The product topology on the Cartesian product λ∈Λ Xλ is the Q topology that has as a base sets of the form λ∈Λ Oλ where each Oλ ∈ Tλ is open and Oλ = Xλ except for finitely many λ. ...
The non-Archimedian Laplace Transform
... basis of our investigations are the results in the general theory of locally convex spaces [9 - 14]. Then we have proved that the Laplace transform is a topological isomorphism (this fact is very useful to prove that solutions of differential equations depend continuously on initial data). The resul ...
... basis of our investigations are the results in the general theory of locally convex spaces [9 - 14]. Then we have proved that the Laplace transform is a topological isomorphism (this fact is very useful to prove that solutions of differential equations depend continuously on initial data). The resul ...
