What is a Hamel Basis and How is it Used in Vector Spaces?

[Sourabh N]

I learned that given a vector space we can construct a basis for it. Can anyone give me an example of this thing (except the trivial example of 2-D and 3-D euclidean space.)

 

[mathwonk]

well a basis for the space of solutions of the differential equation (D-a)^n f = 0, is given by the functions e^(ax), xe^(ax),...,x^(n-1)e^ax).

hows that?or just find a basis for the solutions of the equation x+y+z = 0.

 

[Johan de Vries]

Hmmm, can we always construct a basis? I thought that one needs to invoke the Axiom of Choice to show that every vector space has a basis...

 

[Sourabh N]

First of all, thanks mathwonk for that example.
@ Johan de Vries : As far as I know, for a finite dimensional space we can always construct a basis (though not uniquely).

 

[Sourabh N]

One basis for sols of x+y+z=0 is { (1,0,-1), (0,-1,1) }. This one is orthogonal. But what is the standard basis for this vector space?

 

[radou]

First of all, thanks mathwonk for that example.
@ Johan de Vries : As far as I know, for a finite dimensional space we can always construct a basis (though not uniquely).

Correct.

One basis for sols of x+y+z=0 is { (1,0,-1), (0,-1,1) }. This one is orthogonal. But what is the standard basis for this vector space?

What do you mean by "standard"?

 

[Sourabh N]

In my prof's language "God Given".

This vector space is isomorphic to R^{2}, I suppose. So, do I have a standard basis here as I had there.
http://en.wikipedia.org/wiki/Standard_basis

 

[Johan de Vries]

Yes, for finite dimensional vector spaces you can always construct a basis. However for infinite dimensional vector spaces you can only show that a basis exists. Here basis means "Hamel Basis", i.e. a set of vectors such that every element is a unique finite linear combination of the elements of the set.

 

[radou]

In my prof's language "God Given".

This vector space is isomorphic to R^{2}, I suppose. So, do I have a standard basis here as I had there.
http://en.wikipedia.org/wiki/Standard_basis

Yes, it is isomorphic to R^2, since their dimensions are equal.

I'm not 100% sure about this, but I think the term "standard basis" referrs only to widely known and frequently used vector spaces, such as R^n, C^n, etc.

 

[morphism]

Yes, for finite dimensional vector spaces you can always construct a basis. However for infinite dimensional vector spaces you can only show that a basis exists.

Careful with the wording. You probably meant to say we can't construct a basis for the 'general' infinite dimensional vector space. Because we can construct bases for many specific examples, e.g. {1, x, x^2, x^3, ...} is a basis for the space of polynomials in x (over a fixed field).

 

[mathwonk]

IF you can find a finite spanning set, then you can change it into a basis. there is to my knowledge no constructive way to actually find a finite spanning set for a given abstract space.

i.e. given an infinite spanning set, there is no finite construction to reduce it to a finite spanning set.

so strictly speaking it is not true that for an abstract finite dimensional space one can construct a basis.

 

[leon1127]

indicator function on L2 is an example of basis

 

[Sourabh N]

Sorry, but what is L2 , indicator function?

 

[HallsofIvy]

indicator function on L2 is an example of basis

Yes, but that is irrelevant to the question, "how does one construct a basis for a given vector space?"

 

[mathwonk]

as i said, in general it is not trye that thgere is a constructive way to porudce a basis of a finite dimensional space, as there is no constructive way to "choose a vector not in the span of the ones already chosen", as this argument is sometimes phrased.

 

[mathwonk]

and who is hamel by the way? a hamel basis is nothing but a plain old basis as far as i can see.

according to wiki some people think "hilbert bases" are bases, hence use hamel to reinforce the fact that they are talking about vector bases. this must be very archaic practice as no one has ever used this terminology to me in over 50 years of study and research.

in fact never having eard hamel said out loud i do not even know the correct pronunciation. this is a term one encounters only in reading old sources.

probably the modern term would be just basis or vector basis as opposed to hilbert basis or maybe schauder basis.

 

[morphism]

I've always believed that they're called Hamel bases because Hamel tried to prove the existence of a discontinuous linear map by considering R as a vector space over Q.

 

[Johan de Vries]

Physics students will first encounter Hilbert spaces and Hilbert bases in quantum mechanics courses. Only later will they study functional analysis (if at all). When I studied functional analysis the Prof. did mention Hamel Basis, perhaps because he knew that there were a few physics students like me following the course :)

I have forgotten most of this functional analysis stuff, b.t.w. I only remember the Hahn-Banach theorem which we proved using transfinite induction in the course.

 

[mathwonk]

speaking of category theory, i always remember the hb theorem as saying that C is an injective object in the category of banach spaces?

and how could it haVE BEEN A serious problem to construct a discontinuous (i.e. unbounded) linear map?

 

[mathwonk]

wolfram says hamel basis refers to the specific example of a Q vector basis of R.