I'm taking a functional analysis course. I've already been working on orthonormal sets and I'm currently learning about total sets. The definition is from Kreyszig [1]: https://i.stack.imgur.com/AOgxh.png. I want to solve this problem:
Let $(e_n)_n$ a total orthonormal sequence in a Hilbert space H. i. Prove that ${(e_n-e_{n+1})}_n$ is a total sequence. ii. Is the previous result true as well if $(e_n)_n$ is just a total sequence?
I want to use the following Lema (3.3-7) to solve it: "For any subset $M\neq{\emptyset}$ of a Hilbert space H, the span of M is dense in H if and only if $M^{\bot}=\left\lbrace 0\right\rbrace $". I don't know how to do that.
If $X$ is orthogonal to $e_n-e_{n+1}$ for all $n$ then $ \langle x, e_n \rangle=\langle x, e_{n+1} \rangle$ for all $n$. Since $\sum |\langle x, e_n \rangle|^{2}\leq \|x\|^{2}<\infty$ and all the terms of this sequnece are equal it follows that $\langle x, e_n \rangle=0$ for al $n$. Hence, $x=0$. This proves the first part.
Consider $(x^{n})$ in $L^{2}[0,1]$. This is total by Weierstrass Theorem and the fact that any fucntion in $L^{2}[0,1]$ can be approximated by continuous fucntions. But $(x^{n}-x^{n+1})$ is not total since all these functions vanish at $x=1$.