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By Frithjof Nevanlinna, Rolf Nevanlinna

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Now let B be an arbitrary real symmetric bilinear function on R"' which does not vanish identically. Then if Bx' assumes positive values, § 4. Bilinear and Quadratic Functions 33 for example, let Ut be a maximal positive space, that is, a subspace of the highest possible dimension p where B is positive definite. In this subspace we can by means of the Schmidt orthogonalization procedure construct a coordinate system which is orthonormal with respect to B. ,eP). We are now going to project the R"-vector x onto U' and determine the corresponding normal of x to UP.

4. Bilinear and Quadratic Functions 31 On the other hand, the positive space Ut and the negative space IM are in general not invariant as subspaces, however their dimensions p and q are. To see this, we consider a second decomposition of the required kind, RX, = UP + VQ + W' , where by the above W' = W'. Let the dimensions of L1 and V be p and q; the claim is that p = p and q = q. In fact, for an arbitrary u in UP one has uniquely u=u+v+ v by virtue of the second decomposition. Here u = A u is obviously a linear mapping of the space U' into the space V "P, and indeed a regular mapping.

It is because of the existence of this natural topology that the basic notions and relations in the "absolute analysis" developed in the following chapters will be for the greatest part of a purely affine character in finite dimensional spaces. We shall, however, partly in order to be able to formulate the concepts and theorems conveniently and partly for technical reasons having to do with the proofs, almost everywhere use the metric concepts introduced in this section. But the notions and theorems of absolute analysis in themselves will mostly be independent of the particular choice of the auxiliary metrics introduced.

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