Hilbert Space

Hilbert spaceThat is, the complete inner product space, which can be understood as a complete vector space with inner products.

Hilbert space is based on finite-dimensional Euclidean space and can be seen as a generalization of the latter. It is not limited to real numbers and finite dimensions, but it is not complete. Like Euclidean space, Hilbert space is also an inner product space, and has the concepts of distance and angle. It is also a complete space, and all Cauchy sequences on it converge to one point, so most concepts in calculus can be extended to Hilbert space without obstacles.

Hilbert space provides an effective way to express Fourier series and Fourier transform based on polynomials in any orthogonal system. This is one of the core concepts of functional analysis, and it is also one of the key concepts of postulate mathematics and quantum mechanics.