ABSTRACT
This chapter discusses a brief and basic introduction to the part of the functional analysis dealing with function spaces. Thus, as a rule, main theorems are introduced without proofs, which can be found in the extensive mathematical literature. One of the central questions of the spectral theory is the property of completeness of the system of eigenfunctions in the linear space under consideration. The infinite dimensional spaces, that is, the spaces having an infinite number of linearly independent vectors, play the main role in the functional analysis. The concept of embedding of linear spaces is helpful in situations when one is using several spaces at the same time. A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses.
