$l^{p}$ is not finite dimensional

Question

Well, the exercise was to prove that $l^{p}$ is not finite dimensional space for $p$=2.

I did it proving that the unit ball is not compact. Easy.

However, i was trying to build an element $x \in l^{p}$ that given a finite basis $\lbrace e_{1}, ... ,e^{n} \rbrace$ cannot be obtained via finite linear combinations of elements of basis. Probably this is quite easy, but i couldnt. Can you help me ?

Thank you :)

Answer 1

It's easy to write down an infinite set of linearly independent vectors. Just let $v_n$ be the sequence which has $1$ in the $n$th place and $0$ everywhere else.