ABSTRACT
This chapter explores random functions in abstract metric space, expectation and covariance in a Hilbert space, Gaussian random functions and principal component analysis. It provides a concise introduction to the most important concepts of theoretical FDA. In order to extend the concepts of the expected value and variance, and establish extensions of results like the law of large numbers and the central limit theorem, one must assume that the random elements take values in a vector space. A number of profound results can be established in Banach spaces assuming they have some additional properties. One can show that every covariance operator is Hilbert-Schmidt. All random functions take values in a separable Hilbert space. Just as in the scalar case, the expected value and the covariance operator do not determine the distribution. The appropriate extension of the characteristic function is the characteristic functional.
