Date: Mar 18, 2013 1:28 AM
Author: fom
Subject: Re: Matheology § 224

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.