Ernst Snapper

Abstract: natural to introduce a metric in the space of linear transformations and to call that transformation the most economical which is closest to the identity. Again, it is natural to use a metric that is intrinsically con nected with the two given forms. A positive definite form represents a metric in the vector space. In a well known manner one can derive from it a metric in an arbitrary tensor space, especially in the space of linear transformations. It is remarkable that the two minimum problems derived in this way from the given forms have the same unique solution. Its matrix T 0 can be expressed by the matrices A and B of the given forms as that square root {A~ x B) l l 2 whose characteristic values are all positive. These elementary considerations can be generalized to Hubert space by a suitable modifica tion of the extremum problem. It can be shown that a suitable (A^B) 112 has similar extremum properties as in the finite-dimensional case. lists several definitions of the word "surface n and uses recent results in the field to show how the term "area" can be applied to each. The principal result is that in each case the area is a lower semi-continuous function of the surface. (Re ceived October 17, 1945.) family only of family the set. of