Complements in Functional Analysis

Functional analysis deals with properties of spaces of functions (or of sequences), rather than of individual objects. To study a specific function is doing real (or complex, or more generally hypercomplex) analysis. To study a vector space of all functions with a given common property pertains to functional analysis and uses the fact that this space may be a Hilbert, Banach, or Fréchet space, or have another structure. This structure will determine notions such as convergence of sequences and characterization of continuous functionals on the given space.