- #1

cianfa72

- 2,111

- 235

- TL;DR Summary
- 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.

Last edited: