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,
we can replace that sequence with a trailing sequence of ones,
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
we can associate 1 with the constant sequence,
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
may be representationally equivalent to
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...
1111.. 11111... 110111... 1101111...
whose coordinatewise product is
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