# Span of an infinite set

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

Staff Emeritus
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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.

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.