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Some New Characterization in Barrelled Spaces Wutan Huatan Jisuan Jishu ISSN:1001-1749 Some New Characterization in Barrelled Spaces (1)Dr. Pramod Kumar Assistant Professor (Contract) Katihar Eng. College, Katihar (2)Dr. Rajnish Kumar S.K. Mahila College, Jehanabad Introduction : The Basic definition of a barrel "A locally convex space E is a subset of E which is absolutely convex, absorbent and closed". Here, we go the other way-round. We take the necessary and sufficient condition for a barrel given in (a) below as definition and give new characterization to the theory of Barrelled spaces. (a) Let E be a separated convex space with dual E'. A subset B of E is called a barrel, iff B is the polar of a E',E -bounded subset A of E'. Key words : Topological vector space, locally convex space, Barrell, -Barrel, Barrelled space, Non locally convex space, Countably Barrelled space. Some Basic Definition : i. Topological Vector Space – A set E on which a structure of vector space over K and a topology are defined is a topological vector space if a. The map x, y x y from E × E into E is continuous. b. The map ,x x from K × E into E is continuous. ii. Locally convex space – A topological vector space E is said to be locally convex topological vector space or simply locally convex space or a convex space, if there is a fundamental system of convex nhds. of the origin in E iii. Barrel – In a topological vector space a balanced, convex, closed and absorbing set is called a barrel. Volume XVI, Issue X, OCT/2020 Page No:23 Wutan Huatan Jisuan Jishu ISSN:1001-1749 Let E be a separate convex space with dual E'. A subset B of E is called a barrel if B is the polar of a E',E -bounded subset A of E'. iv. -Barrel – A -Barrel in a locally convex space E is a barrel which is expressible as a countable intersection of closed and absolutely convex neighburhoods. v. Barrelled space – A locally convex space E is said to be barrelled space, if every E is a neighbourhood of zero. vi. Non locally convex space – A locally convex space E is said to be Non locally convex spaces in which every barrel is a nhd. of zero do exists. Example – lp space where 0 < P < 1. vii. Countably Barrelled space – A locally convex space E is said to be countably barrelled, if every -barrel is neighbourhood of zero. Some New Characterization to the theory of Barrelled Spaces : (1) Let E be a locally convex space. Then a barrel in E is absorbing, balanced, closed and convex set in E. Proof : Let Be be a barrel in E. Then B is the polar of E',E -bounded subset A of E' i.e. B=Ao. Then B = Ao is absorbing. Also B = Ao is absolutely convex and s (E',E)-closed. (2) Every locally convex space X has a fundamental system of neighbourhoods of origin consisting of barrels. Proof : Let X be a locally convex space. Then X has a fundamental system U of convex neighbourhoods of origin. It UÎ u then I {aU : a ³ 1} is a balanced neighbourhoods of origin contained in U. In fact, since the scalar multiplication is continuous so correspoinding to UÎ u there exists a nehghbourhood. V of origin and 0 with x U for and x V . Hence, VU for and Volume XVI, Issue X, OCT/2020 Page No:24 Wutan Huatan Jisuan Jishu ISSN:1001-1749 So VU for 1. Then, V w I U : 1 Now since scalar multiplication by a non-zero scalar is a homeomorphism so v is a neighbourhood of origin and hence its superset w and 0 1 then 1, x U and so x U for 1. Hence, x w so that w is a balanced neighbourhood of origin and is contained in U. Also w is convex being an intersection of convex sets. Thus there is a fundamental system v of neighbourhoods of origin consisting of balanced convex sets. Consider the closure of the set in v. Now if L v there is a neighbourhood of origin M v with MML . Then if x M x M intersect M and so x M M M L . Then ML . Now since the closure of balanced convex set is balanced and convex it follows that x has a fundamental system of neighbourhoods of origin consisting of balanced, convex, closed and absorbing sets i.e. x has a fundamental system of neighbourhoods of origin consisting of barrels. (3) Let (E, E') be a dual pair. A subset A of E being a barrel in E depends only onth dual pair (E, E'). Proof : The closure of an absolutely convex set in E, it the same for all topologies of the dual pair (E, E'). Hence the result. (4) A necessary and sufficient condition for a separated convex space X with dual X' to be barrelled is that every weakly bonded subset of X' is equicontinuous i.e. that the topology of X is the strong topology. Volume XVI, Issue X, OCT/2020 Page No:25 Wutan Huatan Jisuan Jishu ISSN:1001-1749 Proof : Suppose X is a barrelled space. Then every barrel in X is a neighbourhood of origin. Let M be a weakly bounded subset of X. Then its polar Mo is balanced, convex, closed and obsorbing set in X. Thus Mo is a barrel in X. Hence by supposition Mo is neighbourhood of origion in X. Moreover MMÍ o . Thus M is contained in the polar of neighbourhood of origin. Thus M is equicontinuous. Thus every weakly bounded subset of X' is equicontinuous. Conversely suppose that every weakly bounded subset of X' is equicontinuous. Let U be a barrel in X. Then by the theorem of biopolars we have U = Uo. Now since U is absorbing, Uo is a weakly bounded subset of X'. Hence by supposition Uo is equicontinuous. Hence Uo = U is neighbourhood of origion in X. Thus every barrel in X is a neighbourhood of origin and so X is a barrelled space. (5) If X is a separated barrelled space with dual X' then the closed absolutely convex hull of every weakly compact subset of X' is weakly compact. Proof : Let M be a weakly compact subset of X'. Then its absolutely convex hull M is weakly bounded and hence by (4) it is equicontinuous. Hence its closure and any closed subset are weakly compact. Example : Every complete metrizable locally convex space is barrelled i.e. every Frechet space is barrelled. Verification : Let X be a complete metrizable locally convex space. Since X is metrizable there is a fundamental system (un ) of decreasing neighbourhoods of origin in X. Let M be a barrel in X. If M is not a neighbourhood of origin, there is a sequence (xn ) with xnÎ u n but xn Î/ nM . Then (xn ) is a null sequence and so the set L= { xn } U {0} is compact. Hence its closed convex envelope is compact. Hence it is absorbed by M. Therefore, there exist a > 0 with xn Î a M for all n which is contradiction. Thus every barrel in X is neighbourhood of origin i.e. X is a barrelled space. Volume XVI, Issue X, OCT/2020 Page No:26 Wutan Huatan Jisuan Jishu ISSN:1001-1749 (6) Let E be a locally convex space and E' its dual. A subset M of E' is bounded o for the topology s (E',E) if and only if it is contained in the polar B of a barrel B in E. Proof : Let M be s (E',E)-bounded. Then by definition of a barrel, B=Mo is a barrel o oo o o in E. Now BMM= Ê i.e. MBÍ . Hence M is contained in the polar B of a barrel B in E. Conversely, Let MBÍ o , where B is a barrel in E. By definition of a barrel, o o oo BA= where A is s (E',E)-bounded subset of E'. Now MBAÍ = . But o o oo oo BA= is absorbing then BA= is s (E',E)-bounded. Hence MAÍ is s (E',E)-bounded. REFERENCES : 1. A. Grothnendick : Topological Vector spaces. Springer Verlag, Berlin. 1964. 2. Aharon Atzmon : An operator on a Freehet space with No common invariant subspace with its inverse. Journal of Functional Analysis Vol. 55- 68-77 (1984). 3. A. Mallics : Topological Algebras : Selected Topics (North Holland Mathematics Studies. 124(1986). 4. A. Rodes – Usan : Geometric Aspects of Banach Spaces. London Mathematical Society Lecture Note Series (1989) 5. Horvath. J. : Topological Vector space and Disributions. Addison Wesley, Massachusetts (1966) 6. I.M. James : Introduction to Uniform Spaces London Mathematical Society Lecturer Note Series 144 (1990). 7. J.I. Nagata : Modern general Topology. North-Holland Volume XVI, Issue X, OCT/2020 Page No:27 Wutan Huatan Jisuan Jishu ISSN:1001-1749 Mathematics Library, 33(1985). 8. J. Mujica : Complex Analysis, Functional Analysis and Approximation Theory. North Holland Mathematics Studies, 125(1986) 9. Kothe. G. : Topological Vector Spaces, Vol. I and II. Springer Verlag, berlin, 1969 and 1989 10. Robertson, A.P., and : Topological Vector Spaces. Robertson, W.J. Cambridge University Press. 1964. 11. Schaefer. H.H. : Topological Vector Spaces. Macmillan. 1964. 12. Treves. F. : Topological Vector Spaces. Distributions and Kernels Academy Press. 1987. 13. Wilansky, A. : Topological Vector Space. Springer Verlag, Berlin. 1979. 14. Wolaka. J. : Topological Vector Spaces. 1987 15. Wolfgang Lusky : A Note on Banach Spaces Containing Co and Cv 16. Wong, Yau-Chuen.
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