Let (E,ξ）=indlim (En,ξn） be an inductive limit of a sequence of locally convex spaces,For brevity,denote by (DS) each set Bbounded in (E,ξ） is contained in some En； and (DST) each set B bounded in (E,ξ） is co...Let (E,ξ）=indlim (En,ξn） be an inductive limit of a sequence of locally convex spaces,For brevity,denote by (DS) each set Bbounded in (E,ξ） is contained in some En； and (DST) each set B bounded in (E,ξ） is contained and bounded in some (En,ξn）. Theovem 1.(DS) holds provided that (i) for each n∈N,there is a neighborhood Un of o in (En,ξn） and m(n)∈ such that -↑Un^E包含于Em(n),and (ii) for any neighborhood V n of o in (En,ξn）,∞↑Un=1 Vn absorbs every bounded set in (E,ξ）. theorem 2 Let all (En,ξn） be metrizable and (DS) hold,then for each bounded set B IN (E,ξ）and each n ∈N thcrc is a neighborhood U k of o in (Ek,ξk）, 1≤k≤n ,and m(n)∈N such that ——↑(B+U1+U2+…+Un)^E包含于 Em(n). theorem 3. Let all (En,ξn） be Frechet spaces.Then (DST) holds if and only if (i) for each n ∈N,there is u neighborhood U n of in (En,ξn） and m(n)∈N such that 0↑Un^E包含于Em(n),and (ii) for each each closed ,absosed,absolutely conuex,bounded set B in (E,ξ）,∞↑Un=1(（εnB)∩Un）absorbs B,where U n is any neighborhood of o in (En,ξn） and εn is any positive number for every n ∈N。展开更多
We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding ...We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.展开更多
文摘Let (E,ξ）=indlim (En,ξn） be an inductive limit of a sequence of locally convex spaces,For brevity,denote by (DS) each set Bbounded in (E,ξ） is contained in some En； and (DST) each set B bounded in (E,ξ） is contained and bounded in some (En,ξn）. Theovem 1.(DS) holds provided that (i) for each n∈N,there is a neighborhood Un of o in (En,ξn） and m(n)∈ such that -↑Un^E包含于Em(n),and (ii) for any neighborhood V n of o in (En,ξn）,∞↑Un=1 Vn absorbs every bounded set in (E,ξ）. theorem 2 Let all (En,ξn） be metrizable and (DS) hold,then for each bounded set B IN (E,ξ）and each n ∈N thcrc is a neighborhood U k of o in (Ek,ξk）, 1≤k≤n ,and m(n)∈N such that ——↑(B+U1+U2+…+Un)^E包含于 Em(n). theorem 3. Let all (En,ξn） be Frechet spaces.Then (DST) holds if and only if (i) for each n ∈N,there is u neighborhood U n of in (En,ξn） and m(n)∈N such that 0↑Un^E包含于Em(n),and (ii) for each each closed ,absosed,absolutely conuex,bounded set B in (E,ξ）,∞↑Un=1(（εnB)∩Un）absorbs B,where U n is any neighborhood of o in (En,ξn） and εn is any positive number for every n ∈N。
文摘We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.
