On Mon, May 23, 2005 at 06:03:32PM -0700, "Hal Finney" wrote:
> Paddy Leahy writes:
> > Oops, mea culpa. I said that wrong. What I meant was, what is the 
> > cardinality of the data needed to specify *one* continuous function of the 
> > continuum. E.g. for constant functions it is blatantly aleph-null. 
> > Similarly for any function expressible as a finite-length formula in which 
> > some terms stand for reals.
> I think it's somewhat nonstandard to ask for the "cardinality of the
> data" needed to specify an object.  Usually we ask for the cardinality
> of some set of objects.
> The cardinality of the reals is c.  But the cardinality of the data
> needed to specify a particular real is no more than aleph-null (and
> possibly quite a bit less!).
> In the same way, the cardinality of the set of continuous functions
> is c.  But the cardinality of the data to specify a particular
> continuous function is no more than aleph null.  At least for infinitely
> differentiable ones, you can do as Russell suggests and represent it as
> a Taylor series, which is a countable set of real numbers and can be
> expressed via a countable number of bits.  I'm not sure how to extend
> this result to continuous but non-differentiable functions but I'm pretty
> sure the same thing applies.
> Hal Finney

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!

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.


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A/Prof Russell Standish                  Phone 8308 3119 (mobile)
Mathematics                                    0425 253119 (")
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