All $e_i's$ are in $l_1$ but $e_1+e_2+...+e_n+...$ is not in $l_1$, why? Hint for the the pattern of basis for $l_1$ space.

Question

Let us consider $l_1$ space with the norm $||x||=\sum_{i=1}^\infty|x_i|$ where $x=(x_1,x_2,...,x_n,...)$. Now let us take

$e_1=(1,0,0,...$), $e_2=(0,1,0,0,...$),$...$,$e_n=(0,0,0,...,1,0,0,...)$.

All $e_i's$ are in $l_1$ but $e_1+e_2+...+e_n+...$ is not in $l_1$ while $l_1$ is a vector space so it should be closed under addition. Why is it not closed under addition? Is closure property for only finite addition? Can somebody hint for the pattern of the basis for $l_1$ space? please!

Answer 1

"Closed under addition" means just finite sums. For infinite sums you need limits, as you do in calculus for an infinite series of numbers. In this situation you usually want for a basis a set of vectors for which finite sums are dense. The set of all the $e_i$ is one.

If you want a basis with the usual definition (finite sums of elements) you will need the axiom of choice and the result will be ugly - that is, it won't tell you anything about the norm structure.