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).