Lee Corbin writes:
> 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
> 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.
I don't think that is exactly how the result Russell describes works, but
certainly Lee's construction makes his result somewhat less paradoxical.
Indeed, a real number can include the information from any countable
set of reals.
Nevertheless I'd be curious to see an example of this Hamel basis
construction. Let's consider a simple Euclidean space. A two dimensional
space is just the Euclidean plane, where every point corresponds to
a pair of real numbers (x, y).
We can generalize this to any number of dimensions, including a countably
infinite number of dimensions. In that form each point can be expressed
as (x0, x1, x2, x3, ...). The standard orthonormal basis for this vector
space is b0=(1,0,0,0...), b1=(0,1,0,0...), b2=(0,0,1,0...), ....
With such a basis the point I showed can be expressed as x0*b0+x1*b1+....
I gather from Russell's result that we can create a different, countable
basis such that an arbitrary point can be expressed as only a finite
number of terms. That is pretty surprising.
I have searched online for such a construction without any luck.
The Wikipedia article, http://en.wikipedia.org/wiki/Hamel_basis has an
example of using a Fourier basis to span functions, which requires an
infinite combination of basis vectors and is therefore not a Hamel basis.
They then remark, "Every Hamel basis of this space is much bigger than
this merely countably infinite set of functions." That would seem to
imply, contrary to what Russell writes above, that the Hamel basis is
uncountably infinite in size.
In that case the Hamel basis for the infinite dimensional Euclidean space
can simply be the set of all points in the space, so then each point
can be represented as 1 * the appropriate basis vector. That would be
a disappointingly trivial result. And it would not shed light on the
original question of proving that an arbitrary continuous function can
be represented by a countably infinite number of bits.