A Hamel basis is a set H such that every element of the vector space is a *unique* *finite* linear combination of elements in H.
This can be proven using Zorn's lemma, which is a direct consequence of the Axiom of Choice. The idea of the proof is as follows. If you start with an H that is too small in the sense that some elements of the vector space cannot be written as a finite linear combination of members of H, then you make H a bit larger by including that element. Now H has to satisfy the constraint that any finite linear combination of its elements be unique. Adding the element that could not be written as a linear combination will not make the larger H violate this constraint. You can imagine adding more and more elements until you reach some maximal H that cannot be made larger. The existence of this maximal H is guaranteed by Zorn's lemma. If you now consider the union of H with any element of the vector space not contained in H, then the condition that any finite linear combination be unique must fail (otherwise the maximality of H would be contradicted). From this you can conclude that the element you added to H (which was arbitrary) can be written as a unique linear combination of elements from H. Saibal ------------------------------------------------- Defeat Spammers by launching DDoS attacks on Spam-Websites: http://www.hillscapital.com/antispam/ ----- Oorspronkelijk bericht ----- Van: ""Hal Finney"" <[EMAIL PROTECTED]> Aan: <firstname.lastname@example.org> Verzonden: Tuesday, May 24, 2005 06:07 PM Onderwerp: RE: White Rabbit vs. Tegmark > 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. > > Hal >