MATEMATHQKHBECHHK 54 (2002), 111-115 UDK 517.982.2 oprrr-anaznra HaY'-IHVl pazr research paper ON THE THREE-SPACE-PROBLEM FOR dF SPACES AND THEIR DUALS Zoran Kadelburg and Stojan Radonovic Abstract. It is shown that dF spaces of K. Brauner behave more regularly than DF spaces in connection with the three-space-problem. In particular, this problem has a positive answer in the class of Frechet spaces for the property of being the strong dual of a barrelled dF space. Thus, a partial positive answer to a question of D. Vogt is obtained. 1. Introduction K. Brauner introduced in [4] the class of dF spaces which have some similar properties as more familiar DF spaces of A. Grothendieck, but are also significantly different. Thus, for example, this class is stable under passing to an arbitrary closed subspace, which is not the case for DF spaces. Also, each Frechet space is the strong dual of a dF space, which is not true for DF spaces. In this paper we shall prove some more properties of dF spaces, particularly those connected with the three- space-problem for dF and dual spaces. A Frechet space is usually called a dual space if it is the strong dual of a barrelled DF space. D. Vogt posed the question in [18] whether the property of being a dual space is three-space-stable, i.e., whether if in a short exact sequence (1) of Frechet spaces, F and G are dual spaces, the same must be true for E. It was proved in [6; Examples 1 and 2] that the answer is negative; it is negative also in the class of Banach spaces [5]. In this paper we shall show that the answer to the Vogt's question is positive if the class of dual spaces is replaced with the class of strong duals of barrelled dF spaces. AMS Subject Classification: 46A03, 46A04 Keywords and phrases: Three-space-problem, DF space, dF space, dual Frechet space. Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niska banja, Yugoslavia, October, 2-6, 2002. This research was supported by Ministry of Science, Technology and Development of Serbia, project no. 1856. 111