Having measured D numerical properties of a physical object E which requires many more than D parameters for its complete specification, an observer seeks to estimate P other numerical properties of E. This paper describes how he can proceed when E is adequately described by one member m(E) of a Hilbert space [unk] of possible models of E, when he believes that the Hilbert norm of m(E) is very likely rather smaller than some known number M, and (except for section 6) when all the observed and sought-after properties of E are continuous linear functionals on [unk]. Section 6 treats Frechet-differentiable non-linear functionals. A later paper will reduce unbounded functionals on arbitrary topological linear spaces to the present case.