17 The Digital Functional Analysis

This is a course that many things change completely! The remarks made above for analysis, differential equations, probabilities , topology etc apply here. Two functions are equal only up to an accuracy level. An the functions are defined only up to a resolution. 

To define Dirac's Delta (e.g. at zero) , we need two number systems, at two different resolutions. In the coarser resolution the function seems zero every were, and with value not definable at zero, while at the finer resolution, is not zero, everywhere, and finite at zero with a value larger than the largest number of the system of numbers at the coarser resolution .If  integrating at the finer resolution, it gives a number, existing, in the coarser resolution too and equal to 1. All are simple and there is no need for the twisted functional definition of Schwartz, neither of sequential definition of Shilov! Engineers would recognize in this definition of Dirac's Delta, what always had in mind but was never formulated and defined in mathematics, as the concept of finite system of quantities at a specific resolution, had never before been defined in mathematics

All the functions of  functional space are finite many, and the linear space dimension is finite too! Thus the complications of Unbounded, and bounded operators in Hilbert spaces do not exist here. All arguments become easier, and many new theorems of remarkable power can be proved. Although the functions are finite many, they may be large even for a computer according to the depth for a resolution. But making then so many so as to permit a computer to scan them, is always an optimistic attitude as far as trying to prove a theorem in functional analysis.

The interesting theorems are of different nature in the finite resolution functional analysis. E.g. Instead of proving that any "almost periodic" function is a limit of series of purely periodic functions, the interest here is similar to the Shannon-type theorems, concerning the size of the required information:

How large has to me a base of periodic functions to derive exactly at a resolution, and the visible layer a function? And similar many more questions relating the information at the side of base of functions, and at the side of visible accuracy level and depth of resolution.

The complications of the axiom of choice in set theory , disappear too! Let us  look to the celebrated arguments that proves with the axiom of choice in Euclidean geometry that we can cut a finite spherical ball in to finite many pieces, and to resemble them to make two spherical balls of equal radius with the original. In the light of finite resolutions in geometry, the arguments is essentially equivalent to that we can indeed do that but the derived new spherical balls are of lower resolution (so that the sum of the finite many points of the resulting new balls make in total the points of the original ball in higher resolution!)

Creating 3D mental images with the imagination

The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in  box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality. 

Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is  UTILIZED BY ME AND  is  required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.

Rules for phantasy and drawing of figures.

As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of  P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.