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.