On Linear Spaces Which May Be Rendered Complete Normed Metric Spaces

In this paper, we obtain a characterization of linear spaces which may be normed so as to become complete, linear, normed metric spaces. In this connection, K. Kunugui and M. Fréchet have shown that every metric space S is isometric with a subset of a complete, linear, normed metric space. I t follows from our result that if the cardinal number of 5 is the limit of a denumerable sequence of cardinals, then there is no complete, linear, normed metric space isometric with S. Results on topological spaces which may be rendered linear, normed metric spaces and complete, linear, normed metric spaces have been given by A. Kolmogoroff and B. Z. Vulich. I t will be assumed that the reader is familiar with certain elementary portions of the theory of linear and metric spaces, and with transfinite cardinal and ordinal numbers. Using the generalized continuum hypothesis and normal order theorem, we prove the following: