The formal definition of a Hilbert space establishes that it is a vector space endowed with a positive definite Hermitian form. Consequently, a Hilbert space is a subset of the set of vector spaces and the way to recognize them is to verify that they have such Hermitian form, or in other words, if they have an inner product.

Furthermore, all Hilbert space is isomorphic to the space L ^ 2 or l ^ 2 .

But I'm intrigued with the question. Respectfully , I have seen your publications and I see that you handle very interesting and deep concepts. What is the intention of that question?

Greetings from Venezuela .

Actually Im just trying to get closer to the language used by mathematicians.

For example to understand your answer I will have to see what you mean by Hermitian form (I know what an inner product space is) and I also know what an Hermitian operator is (the vectors may well be the eigenvectors of an Hermitian operator, is that what you mean?).

Further I should probably remind myself of what L 2 is(normalizable vectors?)

and I dont see Cauchy sequences mentioned which I remember seeing when I tried to understand this stuff.

I think all you mean is that it is a vector space, with an inner product and normalizable vectors. Is that correct?

I much more use physics language.

hope it improves in Venezuela. Greetings , juan

The formal definition is a vector space with an inner product such that norm defined by ^2 = |v| , turns the vector space into a complete metric vector space, i.e.,every Cauchy sequence is convergent.

Very loosely, it means that infinite sums of vectors in that space is also in that space. You need some topological requirement in quantum mechanics as you are usually dealing with infinite dimensional spaces and want to think about ever increasing terms in sums of states.

Thanks Andrew, its the completness that was bugging me. That an arbitrary

linear combination of vectors, is still one of the vectors.

Dear Juan,

Firstly I give two equivalent definitions for Hilbert space:

1. A Banach space with the norm defined by a inner product, in the sense that //x//^2=( A Banach space is a complete normed vector space, i.e. every Cauchy sequence is convergent);

2. A vector space endowed with a inner product, complete relatively to the topology defined by the norm generated by the inner product.

The finite dimensional spaces are all Hilbert spaces. All separable infinite dimensional Hilbert space is isomorphic to l^2. As other Hilbert spaces we have L^2, the hilbertian Sobolev spaces and some special Hilbert spaces containing generalized solutions for some PDE(perturbed Laplace equation, diffusion equations and much other).

The Hilbert spaces are useful because in a Hlibert space we have orthogonality, orthonormal bases and projections. So we can better approximate a function if this function is an element of a Hilbert space.

In not hilbertian Banach spaces we have fewer properties. The l^p and L^p spaces with p>2 or p

In Hilbert spaces we have "good" properties which we can not encounter in other mathematical structures.

In mathematics we can expose very different types of spaces. For example we have a topological space with the following property: every sequence is convergent, and converges to any element of the space. Clearly this space is not "good". If every sequence is convergent, introducing the notion of convergence is unuseful.

So, mathematics now are much more than applications in other branches of science and technology.

Clearly, now, without a deep knowledge in mathematics, we can not correctly explain "the theory of everything".

Many scientists that know about Hilbert spaces do not know that Hilberts spaces can only use associative division rings as the number system that defines the value of its inner product. Depending on their dimension these number systems that differ in the way that coordinate systems sequence their members. Further, it is possible that a huge number of separable Hilbert spaces share the same underlying vector space. A normal operator manages a private parameter space in its eigen space that is constituted by the selected version of the number system.

The set of closed subspaces of a separable Hilbert space is an orthomodular lattice. Together these subspaces represent the separable Hilbert space. This means that an orthomodular lattice emerges into a separable Hilbert space. Or otherwise said: Each separable Hilbert space represents an orthomodular lattice. John Baez, Samuel Holland and Maria Pia Solèr have intensively investigated Hilbert spaces.

See: https://golem.ph.utexas.edu/category/2010/12/solers_theorem.html

My latest paper lists a lot of facts, tricks, and possible extensions of Hilbert spaces that you cannot yet find in regular textbooks about these abstract storage media.

See: "The Grand Structure of Physical Reality"; https://vixra.org/abs/2010.0233

If you think about what an inner product is, then you encounter interpretation problems. What stimulates a vector space to construct a map onto itself? It is easy to understand why the value of an inner product can be a real scalar, but why is it possible that this value can be a quaternion? Why selects each Hilbert space only one version of a chosen number system? Why is this version determined by the coordinate systems that sequence the members of the number system? Why differ the characteristics of the eigenspaces of operators of a separable Hilbert space from the characteristics of the eigenspaces of operators of a non-separable Hilbert space? These questions are treated in https://vixra.org/abs/2103.0188

This paper also shows that each Hilbert space owns a private natural parameter space that offers the Hilbert space its private geometric symmetry and its geometric center. It is the reason that each Hilbert space owns a category of natural operators that make the Hilbert space a function space.

You will not find these facts in regular textbooks about Hilbert spaces and you will also not find them on Wikipedia.

These facts also indicate that the Hilbert repository exists, which is a system of Hilbert spaces that share the underlying vector space.

A division albebra means I think that if ab=0 then a or b must be zero. This caracterizes fields where each element which is non zero is invertible. Because if I can invert b, then a=0/b =0. So it is field, real, complex or quaternion.

There are systems where ab=0 and yet neither a nor b is zero. Matrix albebra is full of this.

I have yet to see a proof that such systems are not usefull.

Juan Weisz

Please read https://en.wikipedia.org/wiki/Sol%C3%A8r%27s_theorem

Hilbert spaces were discovered at the turn of the century. Each Hilbert space applies just one version of the number system. This fixes the geometric symmetry and the geometric center of the Hilbert space and supplies it with a private parameter space.

Of course with Hilbert space or vector space, the only operation with closure is addition, so you talk about group. With the orthogonality two non zero quantities

can make zero in the internal product.

I dont see any a priori restriction of the base elements of vectors...sorry but I dont.

I have not see quaternions used for that, but I suppose it is possible.