Any finiteness restriction in the definition of "basis" for a vector space appears incompatible with that in general use elsewhere --- see for example http://en.wikipedia.org/wiki/Hilbert_space#Operators_on_Hilbert_spaces It would be interesting to know what significance attaches to the notion of "dimension" defined in this manner, which at the moment strikes me as inexplicably artificial. WFL On 8/27/12, Dan Asimov <dasimov@earthlink.net> wrote:
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 ) ?
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