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This report concerns an area which is situated half-way between classical numerical analysis and symbolic computation (computer algebra). On the one hand, functions and function space operators constitute explicit objects in the suggested algorithms, on the other hand all relations are eventually projected into some R^ so that only classical computational processes are executed. With a suitable programming system at hand, this projection should naturally be carried out automatically in the spirit of computer algebra; however, it seems that expert human interaction and guidance will be necessary for the achievement of reasonable results. The authors treat two aspects of function space problems: The generation of an approximate solution, and the "validation" of such an approximate solution, i.e., the generation of a function strip about it such that the exact solution lies in this strip. The process is called self-validation because it may run automatically on a digital computer and because the successful completion of the algorithm is equivalent to a mathematical proof of the existence of the exact solution within the strip. The basic ideas are straightforward: The coefficients of approximate solutions in a finite-dimensional subspace are found as solutions of algebraic problems which arise by "rounding" the infinite-dimensional problem into the subspace. The dimension of the approximation may be recursively increased in an iterative defect correction way. The great variety of possible roundings (Taylor rounding, Chebyshev rounding, spline rounding, etc.) offers a high flexibility. The defect-correction formulation is mandatory in the validation process which uses generalizations of classical fixed point theorems: The strip must be mapped into itself by a finite-dimensional interval extension of a mapping whose fixed point is the analytic solution. With a proper choice of the mapping and with sufficient effort, a narrow inclusion of the analytic solution may be obtained. Note that the into-ness of the mapping on a suitably generated strip can be discovered by the computer via the comparison of coefficient intervals. The authors have initiated a direction of research and algorithmic design in numerical analysis which should become increasingly important (e.g., for use on high performance work-stations). They present the fundamental mechanisms of their approach quite clearly with the aid of a good number of well-chosen examples. Unfortunately, they have not always been successful in making the details of their processes transparent; a closer study of the report is made unnecessarily difficult by a notation which is often clumsy and inconsistent, by an excess of detail in some