Does a Linear Combination of Vectors in an Infinite Set Have to Be Finite?

[Bipolarity]

Suppose that some infinite set S spans V. Then this means every vector in V is expressible as some linear combination of the vectors in S. Does this combination have to be finite?

It couldn't be infinite, because that necessarily invokes notions of convergence and norm which do not necessarily apply to an arbitrary vector space?

BiP

 

[Office_Shredder]

That's correct. Assuming you have a well defined notion for them, the set of infinite linear combinations is what's called the completion of the span of S.

 

[mathman]

Suppose that some infinite set S spans V. Then this means every vector in V is expressible as some linear combination of the vectors in S. Does this combination have to be finite?

It couldn't be infinite, because that necessarily invokes notions of convergence and norm which do not necessarily apply to an arbitrary vector space?

BiP

You haven't defined V well enough. If V has a topology, then completeness is meaningful.

Without a topology you need to be very precise in defining "spans". It may mean that every vector in V is expressible by a finite linear combination. Essentially the question is answered by "yes" by definition.

 

[jbunniii]

A bit of terminology may be in order.

For an infinite-dimensional vector space, a Hamel basis refers to a basis in the linear algebra sense: every element of the vector space can be expressed uniquely as a linear combination of a FINITE number of elements of the Hamel basis. Every vector space has a Hamel basis, as a consequence of Zorn's lemma, but in general it's not possible to specify one concretely.

As others have noted, if you want to allow infinite linear combinations, there needs to be a topology involved. For an infinite-dimensional topological vector space, one has the notion of a Schauder basis: http://en.wikipedia.org/wiki/Schauder_basis But not every topological vector space necessarily has such a basis. If you impose more structure, then you can have a guarantee: for example, every Hilbert space has a basis in this sense (orthonormal, even).

 

[blue_raver22]

: Yes, the combination must be finite. The definition of a linear combination states that it is a finite sum of scalar multiples of vectors. In an infinite set, there may be an infinite number of vectors, but the combination used to express any given vector in V must be finite. This is because the concept of convergence and norms are not applicable in this context.