which portion to read to be able to answer this question?

Question

Let $l^2 = \{x = (x_n)| x_n \in \mathbb{R}, \sum_{n=1}^{\infty} x_n^2 < \infty\}$ be the Hilbert space of square summable sequences and let $e_k$ denote the $k^{th}$ co-ordinate vector (with $1$ in $k^{th}$ place, $0$ elsewhere). Which of the following subspaces is NOT dense in $l^2$?

1. span$\{e_1-e_2, e_2-e_3, \ldots\}$

2. span$\{2e_1-e_2, 2e_2-e_3, \ldots\}$

3. span$\{e_1-2e_2, e_2-2e_3, \ldots\}$

4. span$\{e_2, e_3, \ldots\}$

I have knowledge in Analysis , metric spaces , Topology. Still I find this problem unacquainted. Can anyone please tell me which portion to read to be able to answer this question?

Answer 1

This is about Hilbert spaces. It should be covered in any Functional Analysis book.

That said, this concrete question only requires a bit of playing with bases and linear combinations, so it should not be that hard to anyone with a solid linear algebra background.