On 3/18/2013 2:10 AM, Virgil wrote: > In article <mOOdnU1k7dJ5ONvMnZ2dnUVZ_rednZ2d@giganews.com>, > fom <fomJUNK@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_spaces >> >> 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. > > MY points are > > (1) The bijective mapping from the set of binary sequences to the set of > paths of a Complete Infinite Binary Tree, was NOT a linear mapping as it > was originally formulated by WM. > > (2) WM is not competent enough to be able to reformat it correctly . > i.e., to make it an actual and obvious linear mapping. > > (3) It is not all that difficult to create an actual and obvious linear > mapping there for someone who knows something more about linear spaces > that WM does. >
Since you use the word "obvious" I am assuming that I did much too much work attempting to establish a scalar multiplication.
But, I would assume that you would be basing it on the representations as real numbers directly. I tried to avoid that because of WM's finitist claims and the abstract definition of tree paths.