By Frithjof Nevanlinna, Rolf Nevanlinna

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Those vectors y for which Bxy=O holds identically in x obviously form a subspace in R'". Provided the dimension r of this space is positive we say that B x y is r-fold degenerate with respect to y. Thus if B x y is not degenerate with respect to y, it follows from the above identity in x that y = 0. If in an arbitrary coordinate system a, Y=Erlfa,, and hence Bxy=i2 &'r7iBara,=iE+lc,$4of 36 1. Linear Algebra then the fact that B is r-fold degenerate with respect to Y is equivalent with the linear homogeneous system of equations (r= 1.....

Retaining the above hypotheses, let T now be an arbitrary linear transformation. Then for a fixed y, B(T x) y is a linear function of x, and according to the theorem Of Frt chet-Riesz, for each y in Rn there exists, therefore, a unique y* in R"' such that B(T x) y = B x y* identically in x. One verifies at once that y*=T*y is a linear transformation of the space R'", the transformation adjoint to T relative to B. We thus have B(T x) y = B x(T* y) identically in x and y. Provided the nondegenerate bilinear function B x y is in addition symmetric, this identity can bewritten B y(T x) = B(T* y) x or.

1. The natural topology of linear spaces of finite dimension. A linear space of f unite dimension possesses a "natural" topology which goes back to the topology of the real multiplier domain. If al, ... -7 a, i-I then, for example, the limit x -+ xo can be defined by the in limits $i -. $4 (i = 1, ... , in), which are meaningful in the domain of-the real numbers, and, indeed, this can be done in a way that is independent of the choice of a coordinate system: If according to this definition x - xo in one coordinate system, then it follows from the finite, linear formulas for the transformation of the coordinates that the same is the case in all coordinate systems.