Unable to understand this Banach Space example The notion of a "Banach space" seems simple. My textbook says it is a normed vector space that is also complete. Normed, Okay. Vector Space, Okay. Complete, Okay.
An example that was given is $R^n$ with norm $||x|| = |x|$
Now I am confused...What does it mean by "with"? 
Suppose I have a function on $R^2$, $sin(x)$. I take the norm/abs value of it which gives me $|sin(x)|$, I do not see anything magical with this process. Does it mean that $|sin(x)|$ belongs to Banach space but $sin(x)$ does not? This is very confusing, can someone help explain?

A Banacha space $X$ has a norm, or is provided with a norm, ussually denoted by something like $\|\ \|$. You can think of $\|x\|$ as the size of $x$ in $X$. Since there may be different norms, one says: "Let $X$ be such and such space with norm $\|\ \|$ defined as ..."
$\mathbb{R}^n$ is a vector space consisting of vectors $x=(x_1,\dots,x_n)$, not of functions. Neither $\sin x$ nor $|\sin x|$ belong to $\mathbb{R}^2$. You can provide $\mathbb{R}^n$ with many norms. The one that your text denotes by $|x|$ is probably the Euclidean norm
$$
|x|=|(x_1,\dots,x_n)|=\sqrt{|x_1|^2+\dots+|x_n|^2}.
$$
This means that you measure the size of a vector $x=(x_1,\dots,x_n)$ by that expression. Other usual norms are
$$\begin{align}
\|x\|_1&=|x_1|+\dots+|x_n|\\
\|x\|_\infty&=\max(|x_1|,\dots,|x_n|)
\end{align}$$
