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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: summation over arbitrary index set?''
by rspuzio on 2008-09-02 15:09:32 |
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| > For real numbers it only makes sense to consider
> countable index sets.
Except when all but a countable number of them are zero,
in which case we can define their sum to be the sum of
the countably many non-zero elements. This doesn't
contradict anything said above, just a way of generalizing
the definition.
The reason any of this is interesting is because we encounter
situations where such a generalized definition allowing
sums over uncountable sets is useful in certain situations.
For instance, in non-separable Hilbert (or, more generally,
Banach) spaces, we can have an uncountable number of basis
vectors and can express any vector in the space as a sum
over these basis elements if we define sums over uncountable
sets in the way described above.
> Yeah, an entry about this would be nice.. If I have time and
> nobody else does it I'll write it later today.
I already wrote an entry on this topic two years ago:
http://planetmath.org/?op=getobj;from=objects;id=7698 |
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