Comments about "Hilbert space" in Wikipedia
This document contains comments about the article Hilbert space in Wikipedia
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In the last paragraph I explain my own opinion.
Contents
Reflection
Introduction
The article starts with the following sentence.

In mathematics, Hilbert spaces (named for David Hilbert) allow generalizing the methods of linear algebra and calculus from the twodimensional and three dimensional Euclidean spaces to spaces that may have an infinite dimension.

This emphasizes the importance that Hilbert spaces belong to the realm of mathematics i.e. the concept Hilbert space does not belong to physics and does not 'physical' exists.
The same with concepts like infinite dimensions. All physical objects have 3 dimensions.

A Hilbert space is a vector space equipped with an inner product operation, which allows defining a distance function and perpendicularity (known as orthogonality in this context).

Also here: a distance function is something different as a (physical) distance.

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinitedimensional function spaces.

I would not use the term 'arise naturally'. The whole problem is into what extend mathematics can be used to study physical problem. Specific imaginary numbers, which includes Hilbert spaces.








1 Definition and illustration










1.1 Motivating example: Euclidean vector space










1.2 Definition










1.3 Second example: sequence spaces










2 History










3 Examples










3.1 Lebesgue spaces










3.2 Sobolev spaces










3.3 Spaces of holomorphic functions










3.3.1 Hardy spaces










3.3.2 Bergman spaces










4 Applications










4.1 Sturm–Liouville theory










4.2 Partial differential equations










4.3 Ergodic theory










4.4 Fourier analysis










4.5 Quantum mechanics










4.6 Color perception










5 Properties










5.1 Pythagorean identity










5.2 Parallelogram identity and polarization










5.3 Best approximation










5.4 Duality










5.5 Weaklyconvergent sequences










5.6 Banach space properties










6 Operators on Hilbert spaces










6.1 Bounded operators










6.2 Unbounded operators










7 Constructions










7.1 Direct sums










7.2 Tensor products










8 Orthonormal bases










8.1 Sequence spaces










8.2 Bessel's inequality and Parseval's formula










8.3 Hilbert dimension










8.4 Separable spaces










9 Orthogonal complements and projections










10 Spectral theory










11 In popular culture










12. See also
Following is a list with "Comments in Wikipedia" about related subjects
Reflection 1  Physics versus Mathematics.
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Created: 7 January 2022
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