Guest But Posted February 14, 2005 Share Posted February 14, 2005 Dumb question: Is the infinite dimensional vector space (with finite number of non-zero components of course) over a countable field countable? I tried to look it up, but couldn't find anything. I don't need a proof, just maybe a little plausible explanation if possible. Link to comment Share on other sites More sharing options...
matt grime Posted February 14, 2005 Share Posted February 14, 2005 Not necessarily, since you've not said if there is a countable basis. There is no such thing as "the" infinite dimensional vector space. You are asking is [math]\coprod_{a \in \Alpha} F_a[/math] countable if each F_a is countable. That depends on the indexing set \Alpha Link to comment Share on other sites More sharing options...
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