### The Continuous Transformation Ring of Biorthogonal Bases Spaces

Duke Mathematical Journal, Vol. 25 (1958), 365-371.

Received August 27, 1957. This paper is part of a thesis submitted to Northwestern University in partial fulfillment of the Ph.D. degree.

Introduction

This note is concerned with biorthongonal bases spaces. These are dual vector spaces M and N over a division ring D such that there exist equipotent bases {x_{i}} for M and {y_{j}} for N with (x_{i}, y_{j}) = δ_{ij}. We shall refer to the basis {x_{i}} as the good basis. Some of the interest in biorthogonal bases spaces stems from the following theorem proved by Mackey: if [M:D] = [N:D] = ℵ_{0}, then M and N are biorthogonal bases spaces.

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Let L(M) denote the ring of *all* linear transformations on M and F(M) the ring of all finite-valued linear transformations in L(M). It is known that any two-sided ideal I of L(M) consists of all those linear transformations of rank less than some infinite cardinal ℵ_{j}, where ℵ_{j} ≤ dim M. From this it can be proven that L(M)/F(M) is primite without one-sided minimal ideals, further that the quotient of any two two-sided ideals in L(M) has this structure. In this paper we show that exactly the same results hold if M and N are biorthogonal bases spaces. In the final theorem we show that L(M,N) and its ideals are not isomorphic to L(M) and its ideals. In fact, we show that if I_{1} ⊃ I_{2} are two ideals in L(M,N), then I_{1}/I_{2} is not regular while it is known that the corresponding object for L(M) is regular. (The symbols "⊃" and "⊂" are used in the strong sense, excluding equality.)