What does a norm of a polynomial space mean? When talking about polynomial vector space, the following example was provided.

A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n}  a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and $a_{j,k} \neq 0$.
An example of a degree-two polynomial is $||X||^2$ $=x_1^2+x_2^2$

I am conflicted regarding this notation. My understanding was that a norm has outputs in $\mathbb{R}$ whereas this is showing the norm to be a polynomial in itself. 
Am I misunderstanding something here?
 A: $\|X\|$ is not really part of the example. It should probably just have been written

An example of a degree-two polynomial is $x_1^2+x_2^2$.

Writing "$\|X\|=$" in front of it seems just to be intended to reassure the possibly-overwhelmed reader that "this is not a totally new scary thing; remember that you have seen this expression before when defining the Euclidean norm even though you didn't know it was called a polynomial in 2 variables back then".
Feel free to feel vaguely offended by the suggestion that you may be overwhelmed enough to need such assurance.
The source you're quoting doesn't appear to be very carefully written: the condition after the displayed equation doesn't really make sense and should probably read something like

where $a_{j,k}\ne 0$ for at least one pair $(j,k)$ with $j+k=n$.

A: This is just suggestive notation; they are not actually saying this is a norm in the sense of analysis.  They are defining the symbol $\|X\|^2$ to mean a certain polynomial in $X$, namely $x_1^2+x_2^2$.  This notation is presumably chosen because this polynomial is related to the familiar Euclidean norm on $\mathbb{R}^2$.
