Tensor structure applied to the least-squares method, revisited

TLDR: The tensor approach to the least square is a classical case of the Hilbert space approach as mentioned in this paper, and the connection is made to Hilbert spaces by demonstrating that the tensor-based approach to least square adjustment can be seen as a tensor version of Hilbert space.

Abstract: The geometrical approach to the least-squares, based on differential geometry with tensor structure and notations, describes the adjustment theory in a simple and plausible manner. The development relies heavily on orthonormal space and surface vectors, and on the extrinsic properties of surfaces linking the two kinds of vectors. In order to relate geometry to adjustments, the geometrical concepts are extended to an n-dimensional space and u- or r-dimensional surfaces, where n is the number of observations, u is the number of parameters in the parametric method and r is the number of conditions in the condition method, with n=u+r. Connection is made to Hilbert spaces by demonstrating that the tensor approach to the least-squares is a classical case of the Hilbert-space approach.