Russell writes
> You've got me digging out my copy of Kreyszig "Intro to Functional
> Analysis". It turns out that the set of continuous functions on an
> interval C[a,b] form a vector space. By application of Zorn's lemma
> (or equivalently the axiom of choice), every vector space has what is
> called a Hamel basis, namely a linearly independent countable set B
> such that every element in the vector space can be expressed as a
> finite linear combination of elements drawn from the Hamel basis: ie
>
> \forall x\in V, \exists n\in N, b_i\in B, a_i\in F, i=1, ... n :
> x = \sum_i^n a_ib_i
>
> where F is the field (eg real numbers), V the vector space (eg C[a,b]) and B
> the Hamel basis.
>
> Only a finite number of reals is needed to specify an arbitrary
> continuous function!
I can't follow your math, but are you saying the following
in effect?
Any continuous function on R or C, as we know, can be
specified by countably many reals R1, R2, R3, ... But
by a certain mapping trick, I think that I can see how
this could be reduced to *one* real. It depends for its
functioning---as I think your result above depends---
on the fact that each real encodes infinite information.
Suppose that I have a continuous function f that I wish
to encode using one real. I use the trick that shows
that countably many infinite sets are countable (you
know the one: by running back and forth along the diagonals).
Take the digits of R1, and place them in positions
1, 3, 6, 10, 15, 21, ... of the MasterReal, and R2 in positions
2, 4, 7, 11, 16, 22, ... of the MasterReal, R3's digits at
5, 8, 12, 17, 23, ... of the MasterReal, and so on, using
the first free integer position of the gaps that are left
after specification of the positions of the real R(N-1).
So it seems that countably many reals have been packed into
just one. (A slightly more involved example could be produced
for the Complex field.)
Lee
> Actually the theory of Fourier series will tell you how to generate
> any Lebesgue integral function almost everywhere from a countable
> series of cosine functions.
>
> Cheers
>
> --
> *PS: A number of people ask me about the attachment to my email, which
> is of type "application/pgp-signature". Don't worry, it is not a
> virus. It is an electronic signature, that may be used to verify this
> email came from me if you have PGP or GPG installed. Otherwise, you
> may safely ignore this attachment.
>
> ----------------------------------------------------------------------------
> A/Prof Russell Standish Phone 8308 3119 (mobile)
> Mathematics 0425 253119 (")
> UNSW SYDNEY 2052 R.Standish.domain.name.hidden
> Australia http://parallel.hpc.unsw.edu.au/rks
> International prefix +612, Interstate prefix 02
> ----------------------------------------------------------------------------
>
Received on Tue May 24 2005 - 08:39:53 PDT