We present a nonlinear functional analytic framework for modeling and processing fuzzy sets in terms of their membership functions. Let X = {x} denote a universe of discourse, and A, a fuzzy set of elements x in X and membership function μA. First, we formally introduce a class C = {Ã} of attributes Ã, and a judgment criterion J in the definition of μA; and explain the role of such an attribute- and judgment-based membership function μA(Ã, J, x) in the interpretation of A as a value-based or uncertainty-based fuzzy set. Second, for uncertainty-based fuzzy sets, we associate with each attribute à (e.g., old), a corresponding event, also denoted by à (e.g., event of being old), the set C of all such events constituting a completely additive class in an appropriate probability space, x being a random variable, vector, or object induced by this space. On such a basis, we present and discuss the role of the membership function μA as a generalization of the concept of posterior probability P(Ã/x). This allows us to introduce rigorously, on a single platform, both human and machine judgment J in assigning objects to fuzzy sets by minimizing conditional risk. Third, we assume that X is a vector space endowed with a scalar product, such as a Euclidian Space or a separable Hilbert Space. Typically, X = {x} would be a feature space, its elements x = x(p) being feature vectors associated with objects p of interest in the discourse. Then, membership functions become membership “functionals”, i.e., mappings from a vector space to the real line. Fourth, with this motivation, we focus attention on the class Φ of fuzzy sets A whose membership functions μA are analytic (nonlinear) functionals on X, which, can therefore be represented by abstract power series in the elements x of X. Specifically, μA in Φ are assumed to lie in a positive cone Λ of vectors in a Generalized Fock Space F(X). This F(X) is a Reproducing Kernel Hilbert Space (RKHS) of analytic functionals on X introduced by the author and T. A. W. Dwyer in 1980 for nonlinear signals and systems analysis. Fifth, in such a setting, we view the μA as vectors in F(X) acting as “representers” of their respective fuzzy sets A. Thus, because of the one-to-one relationship between the fuzzy sets A and their respective μA, the fuzzy sets A can benefit from all the results of analytical processing of their μA as vectors. Finally, we derive a “best model”  for a fuzzy set A based on a best approximation μA of its membership functional μA in the space F(X), subject to appropriate interpolating or smoothing training data constraints and a positivity constraint. The closed form solution μA thus obtained appears in the form of an artificial network, proposed by the author in another theoretical context in 1990. For the purpose of illustration of the underlying technology, an application to the diagnosis of Alzheimer’s disease is briefly discussed.