In this chapter we recall the basic facts regarding real vector spaces endowed with a scalar product. We introduce the concept of Hilbert space and show that, even for the infinite-dimensional ones, continuous linear functionals are induced by the scalar product. Moreover, we see that even in some classes of infinite dimensional spaces (the so-called separable ones) there exists a well-defined notion of basis (the so-called complete orthonormal systems), obtained replacing finite sums with converging series. Even though the presentation will be self-contained, we assume that the reader has already some familiarity with these concepts (basis, scalar product, representation of linear functionals) in finite-dimensional spaces.
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