The rationale for the restriction to finite sums is that infinite sums are not in general meaningful in a vector space. The question I posed is purely in the category of vector spaces, so no additional structure is relevant for its solution. --Dan P.S. A basis for a vector space V can be defined in either of two equivalent ways: any minimal set of vectors that spans V, or any maximal set of vectors that is linearly independent. The dimension of V is the cardinality of any basis of V; this does not depend on the choice of basis.) On 2012-08-26, at 8:00 PM, Fred lunnon wrote:
But what is the rationale behind the apparently arbitrary restriction to finite sums in your definition of "dimension"? A useful definition should surely be based on considerations of topology, geometry, linear operators, etc.
For example, is it the case that a bounded linear operator in Hilbert space can be represented by a matrix of denumerable dimension ( |N, aleph-zero ) ?