[AntonioLao]

The function space and sequence space of Von Neumann’s theory were described by using an axiomatic approach. There are 5 basic axioms. The first is the existence of a linear vector space. Moreover, addition and scalar multiplication are defined. However, the scalars are assumed to include both real and complex numbers. The inclusion of complex numbers can now be seen as a mathematical attempt to remove the inherent dysfunction of the true absolute vacuum field of space-time quantization. Complex numbers were believed to justify the descriptive transition from discrete space to continuous space, vice versa. However, in a true dysfunctional analysis, complex numbers are not necessary unless the rational insistence of algebraic solution to x + 1 = 0 exists, where x takes on only elements from the real domain. For this reason alone, dysfunctional analysis is the root of all irrational mathematics, not the root of all evils. This implies the existence of many to many correspondences between the domain and range of a normal function.

[Guille]

What are the other four axioms?

[AntonioLao]

Originally Posted by GUILLE What are the other four axioms?

They can be found in volume 2 of his collected works. For biography see http://en.wikipedia.org/wiki/John_von_Neumann. By tomorrow I could provide more about the other four axioms.

[AntonioLao]

Originally Posted by GUILLE What are the other four axioms?

Hopefully this post would shed some light about the other four axioms.
In the late 1920s John von Neumann was stimulated by the use of linear operators in quantum mechanics to present an axiomatic approach to Hilbert space and to operators in Hilbert space as abstract theory of functional analysis. Although his ideas originated from the work of Norbert Wiener, Weyl, and Banach his major contribution was the formulation of a general eigenvalue theory for a large class of linear Hermitian operators. He invented an L space of complex-valued, measurable, and square integrable functions defined on any measurable set M of the complex plane. His axiomatic foundation was proposed as follows:

• H is a linear vector space. That is, there is an addition and scalar multiplication defined on H so that if f1 and f2 are elements of H and a1 and a2 are any complex numbers, then a1f1 + a2f2 is also an element of H.
• There exists on H an inner product or a complex-valued function of any two vectors f and g, denoted by (f, g), with the properties: (i) (af, g) = a(f, g), (ii) (f1 + f2, g) = (f1, g) + (f2, g), (iii) (f, g) = overbar(g, f), (iv) (f, f) ≥ 0, (v) (f, f) = 0 if and only if f = 0.
• In the metric just defined H is separable, that is, there exists in H a countable set dense in H relative to the metric ||f – g||.
• For every positive integer n there exist a set of n linearly independent elements of H.
• H is complete. That is, if {fn} is such that ||fn – fm|| approaches 0 as m and n approach ∞, then there is an f in H such that ||f – fn|| approaches 0 as n approaches ∞.

Reference: Morris Kline, Mathematical Thought: From Ancient to Modern Times, Volume 3, pp 1092-93, Oxford University Press, 1972.

[Guille]

Thanks for the effort and interests on my question. I want to center on the fifth axiom:
H is complete. That is, if {fn} is such that ||fn – fm|| approaches 0 as m and n approach ∞, then there is an f in H such that ||f – fn|| approaches 0 as n approaches ∞.

First of all, this axiom impplies directly that |fn|-|fm| is not equal to ||fn - fm|| true? If so, then the inequality is also true for the parallel substraction ||f – fn||, and this means that |fn|-|f| is always a form of the formula ax^2+bx+c, such that a and c are constant and b ir variable. The number of possible permutations of bracket equation formed is 16 times the numbers of different satisfactory equations there are. Now, if a and c satisfy the equation axc=a-c, it means that there is only 1 possible solution for each numbering of a and c, and thus one solution for b each time. and of course, b would always be from (g+h)(i+j) gi+hj, or what is the same, that H will always be parallel in the plane to either the line x=y or x=-y. This is interesting because in my workings for proving the goldbach conjecture, which I've revised cause I remember I used Hilbert spaces to define the patterns I found in the lattices I developed as laws for the boolean algebra (from the complementary axioms, I conjectured that aV¬a is equal to bV¬b for any ordering a,b which is included a same lattice group).

I've got too much out of the theam, I just wanted to note that there is a big connection between linear operations in QM physics and complex analysis in mathematics, this is interesting because advances in math could mean advances in physics (traditionally it has happened the other way round).

[AntonioLao]

Originally Posted by GUILLE First of all, this axiom impplies directly that |fn|-|fm| is not equal to ||fn - fm|| true?

I have to take a timeout to look into this. Von Neumann's based his ideas on the existence of complex numbers but I am trying to take them out of a theory.