Q:

How do I take the norm of terms in the given sequence?

Let $v$ be the vector space of all sequences x = $(x_{n})_{n=1}^{\infty}$ such that $\displaystyle{\lim_{n \to \infty}} x_{n} = 0$. Define a norm on $v$ by $\Vert \textbf{x} \Vert_{\infty}$ = $\displaystyle{\sup_{n \ge 1}}|x_{n}|$.

If the norm is defined as is, I was wondering how we can take the absolute value of $x_{n}$ if $x_{n}$ is not $1$-dimensional?

A:

If the sequences have values in $\Bbb R$ or $\Bbb C$, then each $x_n$ is just a number. If they have values in some other normed space $X$, you may take $\lvert\bullet \rvert$ to be the norm of $X$. In that case (but technically also in the case $X=\Bbb R$) your $\lVert \bullet\rVert_\infty$ may depend on the norm $\lvert \bullet\rvert$ of $X$.

