On Mar 17, 10:28 pm, fom <fomJ...@nyms.net> wrote: > On 3/17/2013 7:11 PM, Ross A. Finlayson wrote: > > > > > A simple and trivial > > continuous mapping was noted. > > > Regards, > > > Ross Finlayson > > That is not enough Ross. > > By definition, a linear map must satisfy > > f(x+y) = f(x) + f(y) > f(ax) = a*f(x) > > So, the domain must at least have the > structure of a module since it needs > to have an abelian addition of domain > elements and a map from the domain > into itself with a scalar multiplication. > > Furthermore, it is unlikely that one > could take the scalar multiplication > to be the Galois field over two > elements since multiplication by > zero would be the zero vector and > multiplication by one would be > the identity map. > > A morphism with that scalar field > could not reasonably be expected > to have a linear map with a > system of real numbers. > > In order to build a scalar that > could even possibly serve this > purpose, given WM's claims related > to various finite processes, one > would have to invoke compactness > arguments involving completed > infinities. > > For example, for any non-zero > sequence of zeroes and ones > that becomes eventually constant > with a trailing sequence of zeroes, > > 1001101000...... > > we can replace that sequence with > a trailing sequence of ones, > > 1001101111...... > > We want to use these forms because > of the products > > 1*1=1 > 1*0=0 > 0*1=0 > 0*0=0 > > Then, coordinatewise multiplications > along the trailing sequence of ones > retains a trailing sequence of ones. > > In addition, on the interval > > 0<x<=1 > > we can associate 1 with the constant > sequence, > > 111... > > Given these facts, we can now say that > a collection of infinite sequences is > "compactly admissible" if for every > finite collection of those sequences > coordinatewise multiplication yields > a sequence different from one > consisting solely of an initial > segment of zeroes followed by > an initial segment of ones. > > In other words, even though > > 000000111... > > may be representationally > equivalent to > > 000001000... > > for some purposes, compact > admissibility has to ignore > what happens in this conversion. > The situation above is > interpreted as corresponding > with a non-compact set of > sequences. > > Given this, sequences like > > 1000... > 11000... > 110000... > 1101000... > > yield > > 1111.. > 11111... > 110111... > 1101111... > > whose coordinatewise product > is > > 1101111... > > So that the original sequence > is compactly admissible. > > Given a construction along these > lines, one could then think of > compactly admissible collections > as possibly forming a sequence space > as described here > > http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_s... > > Obviously, the compactly admissible > collections are not defined as > converging in the sense of a sequence > of partial sums. > > Equally obviously, I have not done > all the work necessary to decide > whether or not this would work. > > My purpose here is to explain that > the scalar multiplication would > require a construction along these > lines just to even begin to talk > about whether or not WM could > do what Virgil is asking.
I looked to it that a linear mapping would need a vector space over a field. Then basically it was found various magma(s), those being a set equipped with an operation closed in the set, using addition being the integer part of natural addition and multiplication the integer part of natural multiplication. But that is not a field because it lacks distributivity, and multiplicative inverses. Then there's the notion to define addition-1 being the non-integer part of natural addition, and addition-2 being the non-integer part of natural addition, that equals one if the non-integer part is zero, so there are two operations with that are associate, transitive, have inverses in the field and distinct identities, but addition-2 isn't distributive.
So, the question is, if Virgil says there exists a field over [0,1], or the elements of the CIBT or Cantor set, there would be a continuous function f: R_[0,1] <-> R that had a (+) b = f^-1(f(a)+f(b)) and a (*) b = f^-1(f(a)f(b)), and that f(ab) = f(a)f(b) and f(a) + f(b) = f(a +b).
So from an apocryphal comment that there is a linear mapping and thus vector space and field over [0,1], I wonder how Virgil backs this claim, as I well imagine it's not a linear function with f(0) = -oo and f(1) = oo. (And it is.)
Then, about compact admissibility, yes there are general notions that if N and R are compactified it's with points at infinity, then about the form and product you mention, there is not an inverse of the product, and I don't see it defined for all the elements of the CIBT or Cantor set. Please feel free to further explain that.