Hamel Basis

From: Saibal Mitra <smitra.domain.name.hidden>
Date: Tue, 24 May 2005 20:10:08 +0200

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

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----- Oorspronkelijk bericht -----
Van: ""Hal Finney"" <hal.domain.name.hidden>
Aan: <everything-list.domain.name.hidden>
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
>
Received on Tue May 24 2005 - 14:17:32 PDT

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