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• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### 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 two-dimensional 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 infinite-dimensional 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.

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Created: 7 January 2022

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