$$\|x\|_2 := \sqrt{\sum_{i = 1}^{\infty} x_i^2}, \quad\quad \|x\|_{\infty} := \sup_{i \in \mathbb{N}} |x_i|.$$
The space I'm working with is
How can I prove the following?
1) how can I show that these two are well-defined? My suggestion for $\|x\|_2$ is that due to the square root it can't be negative, so it has to be well-defined. But what about the other one?
2) I'm supposed to show that they are not equivalent, but I was assuming that they must be?
The first norm is well-defined because, by definition of $\ell^2$, $\sum_{n=1}^\infty{x_n}^2<\infty$ (and, obviously, it's non-negative). The second one is well-defined because, since $\sum_{n=1}^\infty{x_n}^2$ converges, the sequence $\bigl({x_n}^2\bigr)_{n\in\mathbb N}$ converges and therefore the sequence $\bigl(|x_n|\bigr)_{n\in\mathbb N}$ is bounded.
They are not equivalent because if $X_n\in\ell^2$ is such that the first $n$ terms of $X_n$ are equal to $1$ and all others are equal to $0$, then $\|X_n\|_2=\sqrt n$ and $\|X_n\|_\infty=1$.