With this chapter we begin the study of functional analysis, which represents the second mainÂ topic of this book. Just like in the first part of the book we have shown how to extend to an abstract environment fundamental analytical notions such as the integral of a real function, we now intend to explain how to generalize basic concepts from geometry and linear algebra to vector spaces with certain additional structures. We shall first examine Hilbert spaces, where the notion of orthogonal vectors can be defined thanks to the presence of a scalar product. In the next chapter, our analysis will move to the more general class of Banach spaces, where orthogonality no longer makes sense. One could go even further and consider topological vector spaces, but such a level of generality would exceed the scopes of this monograph.
Cannarsa, P., & D’Aprile, T. (2015). Hilbert spaces. In UNITEXT - La Matematica per il 3 piu 2 (Vol. 89, pp. 133–166). Springer-Verlag Italia s.r.l. https://doi.org/10.1007/978-3-319-17019-0_5