ABSTRACT
Most FDA tools make heavy use of Hilbert space theory, and thus it is important to gain comfort using some tools from functional analysis, a field of mathematics that studies objects in abstract vector spaces, rather than in the Euclidean space. This chapter presents basic definitions and facts of Hilbert space theory which have been extensively used in functional data analysis. The monograph of Akhiezier and Glazman (1993) is an extensive survey of the classical theory. A vector space is defined over a field of scalars. When examining covariance operators, it is convenient to work with tensor products and spaces. In finite dimensional spaces, use of tensors can be avoided by stacking matrices in the right way. Such an approach does not work in infinite dimensional spaces and thus use of tensors cannot be avoided if one wants to work with general Hilbert spaces.
