A Note on Linear Topological Spaces*

A space T is called a linear topological space if (1) T forms a linear f space under operations x+y and ax, where x,yeT and a is a real number, (2) T is a Hausdorff topological space,J (3) the fundamental operations x+y and ax are continuous with respect to the Hausdorff topology. The study § of such spaces was begun by A. Kolmogoroff (cf. [4]. Kolmogoroff's definition of a linear topological space is equivalent to that just given). Kolmogoroff calls a set 5 c T bounded, if for any sequence xveS and any real sequence av converging to 0 we have lim,,.^ avxv = 0, where 6 is the zero element of T. He then shows that a linear topological space T reduces to a linear normed space|| if and only if there exists in T an open set which is both convex^ and bounded. In this note, the characterization of other types of spaces among the class of linear topological spaces is studied. Spaces which are locally bounded, that is spaces containing a bounded open set, are found to be "pseudo-normed" on the one hand, and metrizable on the other, but not in general normed. Fréchet «paces, or spaces of type (F), are characterized. The main result of the paper is that a linear topological space T is finite dimensional, and hence linearly homeomorphic to a finite dimensional euclidean space, if and only if T contains a compact, open set. This of course is a generalization to linear topological spaces of the well known theorem of F. Riesz for the space of continuous functions. We first give some needed properties of bounded sets. The following notations will be used throughout. We denote by aS the set of all ax with xeS; by x + 5, the set of all elements x+y where y ranges over S; by S1 + S2, the set of all x+y with xeSi, yeSz.