In the four paragraphs of this chapter we present some basic definitions and facts from functional analysis, as well as applications in special spaces. These preliminaries will be used in the subsequent chapters. Since many introductions to functional analysis are now available, in order to save space, we omit proofs of all of the lemmas, theorems and corollaries given here. Moreover, one will find here only the working tools which are really needed for the development of the theory of bases. We begin by defining various abstract spaces, and we list their most important properties. Then we investigate linear transformations of one space into another, continue with some facts on conjugate spaces, and conclude with results for several spacial spaces.
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