In this paper, we develop a new learning approach, the Hilbert learning. This approach is similar to Fractal learning, but the Fractal part is replaced by Hilbert space. Like the Fractal learning, the first stage is to encode an image to a small vector in the internal space of a learning system. The next stage is to quantize the internal parameter space. The internal space of a Hilbert learning system is defined as follows: A pattern can be interpreted as a representation of a vector in a Hilbert space. Any vectors in a Hilbert space can be expanded. If a vector happens to be in a subspace of a Hilbert space where the dimension L of the subspace is low (order of 10), the vector can be specified by its norm, an L-vector, and the Hermitian operator which spans the Hilbert space. This establishes a mapping from an image space to the internal space P. This mapping converts an input image to a 4-tuple: t (epsilon) P equals (Norm, T, N, L-vector), where T is an operator parameter space, N is a set of integers which specifies the boundary condition. The encoding is implemented by mapping an input pattern into a point in its internal space. We assume that a system uses local search algorithm, i.e., the system adjusts its internal data locally. The search is first conducted for an operator in a parameter space of operators, then an error function (delta) (t) is computed. The algorithm stops at a local minimum of (delta) (t). Finally, the input training set divides the internal space by a quantization procedure.