（LF）-空间的完备性

设（E，ξ）=indlim（E．，ξn）为Frechel空间序列的诱导权限，我们证明了：若每个（Enξ|En）正速完备，则（E，ξ）=indlim（En，ξn）为完备，进而证明了；若对于每个自然数n，存在自然数m（n）使对于任意p≥m（n）有En；则（E，ξ）为完备．     

Remarks on regular (LF)-spaces

Developing ideas found in a recent paper of Gilsdorf to arbitrary topological vector spaces (tvs) one shows that a Hausdorff (LB) _tv -spaceE is regular provided every null-sequence inE has a series convergent subsequence inE. Received by the editors 1980Mathematics Subject Classification (1985 Revision) 46A12.     

Weighted (LF)-spaces of Continuous Functions

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A note on biduals of strict (LF)-spaces

Grothendieck asked in 1954 in [1] the following questions. (1) Is the bidual of a strict inductive limit of a sequence of locally convex spaces the inductive limit of the biduals? (2) Is the bidual of a strict (LF)-space again an (LF)-space? (3) Is the bidual of a strict (LF)-space complete? M. Valdivia gave a (negative) answer to the first question in 1979 in [5]. Since his counterexample is not an (LF)-space, problem (2) remained open. The aim of this note is to present a negative solution to questions (2) and (3). The answer to question (2) is negative even if every step of the (LF)-space is distinguished, in which case the strong bidual is complete by a result of Grothendieck. Moreover, we show that the strong dual of a strict (LF)-space need not be countably barrelled.     

On extension of isometries in (F)-spaces

An (F)-spaceE is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some>0 such thatD(A/2) <D(A) for every subsetA ofE with 0<D(A), whereD(A) denotes the diameter ofA. Our main result goes as follow: LetE be an Imp-constricted (F)-space andU an open connected subset ofE. Assume thatT:U F is an isometry (i.e., a distance-preserving map) which mapsU onto an open subset of the (F)-spaceF. ThenT can be extended to an affine homeomorphism fromE toF. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.     

(a)-spaces and selectively (a)-spaces from almost disjoint families

We investigate a selective version of property (a) and prove a number of results showing that, under certain set theoretical conditions, (a) spaces and selectively (a) spaces behave in a very similar way, at least for separable spaces. Several results regarding the presence of the referred selective version in spaces from almost disjoint families are established; in particular, we give a combinatorial characterization of such presence. Consistent set theoretical hypotheses implying equivalence between being (a) and being selectively (a) within the referred class are presented, as well as hypotheses implying non-equivalence. We also show that the Continuum Hypothesis is independent of the statement asserting the above mentioned equivalence. The paper finishes by presenting some notes and questions on the role of set theoretical assumptions in the subject.     

On the primarity ofH ∞-spaces

It is proved that the poly-discH ^∞ spacesH ^∞(D ^m ) are primary. We use the representation of these spaces as the direct sum of certain polynomial spaces and some properties of the Fourier transforms ofL ¹(π ^m ) functions.