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 , locally convex space, Barrell,  -Barrel, , Non locally convex space, Countably Barrelled space.

Some Basic Definition :

i. – A 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 is called a barrel.

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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 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

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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 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.

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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 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.

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(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.

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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

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