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cianfa72
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- About the definition of vector space in case of infinite dimension
Hi, a doubt about the definition of vector space.
Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled.
Now the point is: if one consider a sum of infinite polynomials (either countable or not) then the result might not be a polynomial at all !
I'm aware of the above "issue" boils down to the "completeness" of the metric space built over the vector space.
Does that means there is actually a "restriction" in the definition of vector space (like one must consider only finite sums of elements in the set) ?
Thank you.
Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled.
Now the point is: if one consider a sum of infinite polynomials (either countable or not) then the result might not be a polynomial at all !
I'm aware of the above "issue" boils down to the "completeness" of the metric space built over the vector space.
Does that means there is actually a "restriction" in the definition of vector space (like one must consider only finite sums of elements in the set) ?
Thank you.
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