In article <email@example.com>, "Ross A. Finlayson" <firstname.lastname@example.org> wrote:
> 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).
A linear mapping between linear spaces need not be in any sense a continuous mapping or involve any continuity at all, as no topological structure is required of linear spaces in general.
So Ross is, as usual, off on an irrelevant tangent again. --