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On Some Topological Vector Spaces Related to the General Open Mapping Theorem

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On Some Topological Vector Spaces Related to the General Open Mapping Theorem Acta Math. Hung. 61 (8-4) (1993), 235-240. ON SOME TOPOLOGICAL VECTOR SPACES RELATED TO THE GENERAL OPEN MAPPING THEOREM L. M. SANCHEZ RUIZ (Valencia) 1. Introduction The spaces Vr, Wr, V and W were introduced by Valdivia [4], showing they were maximal classes of locally convex spaces for the isomorphism and homomorphism theorem, respectively, when in the range a class of locally convex spaces containing barrelled spaces was desired. Mas [2] gave new characterizations of these classes of spaces without using the dual spaces, showing that a locally convex space E(T) is a V~- space if, and only if, there is no locally convex and barrelled topology on E, strictly coarser than T; and E(T) is a V-space, if, and only if, every separated quotient of E(T) is a V~-space. This allowed him to find easy proofs of Valdivia's isomorphism and homomorphism theorem and generalizations to classes of topological spaces. In this way we shall say that a topological vector space E(T) is an E - V~-space if there is no Hausdorff E-barrelled linear topology on E, strictly coarser than T. Moreover, a topological vector space E(:T) is an Z:-V-space if every separated quotient of E(7") is an E-Vr- space. In this paper we introduce the E-strongly almost open mappings and characterize E - Vr- and s - V-spaces in an analogous way as Valdivia [5] characterizes the V~- and V-spaces by means of the strongly almost open mappings. We also obtain some results concerning Z:- B~-complete and s complete spaces, analogous to those obtained by Valdivia [6] in the locally convex case though with different proofs since we are not able to use the dual spaces. Besides, we prove that if a topological vector space E contains an E - V~-subspace (E - V-subspace) of finite codimension, then E is an E - V,-space (s - V-space) and obtain two open mapping theorems. Topological vector space will stand for Hausdorff topological vector space and our notation follows [1]. However, let us recall that if b/and l~ are the sets of all the neighbourhoods of the origin of the topological vector spaces E and F, respectively, and f is a linear mapping of E in F, then f is called weakly singular if N(f) = M{f:l(Y): V E l~}, where N(f) denotes the kernel of f. f is called almost continuous if for each V C P, f-l(V) C E L/. Being f onto, f is called almost open if for each U E U, f(U) E Y. A 236 L.M. SANCHEZ RUIZ topological vector space F is called g-Br-complete if each almost continuous linear mapping with closed graph of each topological vector space E in F is continuous. A topological vector space E is called g - B-complete if each weakly singular almost open linear mapping of E onto each topological vector space F is open and this holds if, and only if, every separated quotient of E is g - Br-complete. If E(T0) is a topological vector space and T is a further linear topology on E, T00~r will denote the linear topology on E, whose neighbourhoods of the origin are the T-closures, ~:r of the T0- neighbourhoods, U, of the origin. 2. /:-strongly almost continuous linear mappings DEFINITION 1. Let E and F be two topological vector spaces and f a linear mapping of E in F. We shall say that f is g-strongly almost open if for each closed string/4= (Un)n~__l in E, f--~f(E) -" ~kgk n/ f(E)~)n=l is a topological string in f(E). THEOREM 1. A topological vector space E(T) is an g- Vr-space if, and only if, each continuous one-to-one g-strongly almost open linear mapping of E onto each topological vector space is an isomorphism. PROOF. Let f be a continuous one-to-one/:-strongly almost open linear mapping of E onto the topological vector space F and suppose f is not an isomorphism. Let V ( n),=l be a closed string in F. As f-l(v) = ~~f-irvn~ ~ JJ,~=l is a closed string in E, f(f-i(V)) F = V is a topological string in F. Hence F is g-barrelled and E is not an g - Vr-space. Conversely, if E is not an g- Vr-space then there exists a Hausdorfflinear topology T* on E, strictly coarser than T, such that E(T*) is/:-barrelled. Now the identity i: E(T) ~ E(T*) is a bijective continuous linear mapping but it is not an isomorphism. However, i is g-strongly almost open since if $ /4 = ( U -)~=1~ is a closed string in E(T) then i(/4) :r = ~:r* is a closed string and, consequently, topological in E(T*). [] THEOREM 2. A topological vector space E(T) is an ~. - V-space if, and only if~ each continuous/:-strongly almost open linear mapping of E in each topological vector space is a homomorphism. PROOF. Let f be a continuous g-strongly almost open linear mapping of E in the topological vector space F. Without loss of generality we may assume f(E) = F. The linear mapping f of the quotient space E/f-l(O) onto F defined by passing to the quotient is one-to-one, continuous and/:- strongly almost open. Hence f is an isomorphism and f is a homomorphism. Conversely, if E is not an /: - V-space then there exists a Hausdorff linear topology T* on ElF, strictly coarser than the quotient topology such Ac~a Mathemalica Hungarlca 61, 1993 .
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