is L a operator (function), a set or a subspace when talk about null space L(v) = 0? per wiki, In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W

is the set of all elements v of V for which L(v) = 0, where 0 denotes
  the zero vector in W.

null space is a set, denoted by L, so, L is a set, in the mean while L(v) = 0 use L as an operator (function), this confuses me.
can anyone give an explanation?
 A: In your example, $L$ is being used to denote a linear map from one vector space ($V$) to another ($W$), not the kernel. The kernel of $L$ is the set of all $v\in V$ such that $L(v)=0$.
For a concrete example, let $V=W=\mathbb{R}$, and let $L:V\rightarrow W, L(v) =(v+2)(v-1)$. Then the kernel of $L$ is the set $\{-2,1\}\subset V$, as $L(-2) = 0$ and $L(1)=0$.
For another example, let $V=W=\mathbb{R}^2$. In some fixed basis, let $L:V\rightarrow W$ where
$$ L = \begin{pmatrix} 
1&2\\2&4
\end{pmatrix}$$
Note that the rank of $L$ is one, so we expect the dimension of the null space to be one as well. We also have that
$$L\left( \begin{pmatrix}-2\\ 1 \end{pmatrix} \right) =\begin{pmatrix} 
1&2\\2&4
\end{pmatrix}\begin{pmatrix}-2\\ 1 \end{pmatrix}  = \begin{pmatrix} 0\\0\end{pmatrix} $$
So with this, we know that the null space is spanned by $\begin{pmatrix}-2\\1\end{pmatrix}$ and thus that 
$$\ker (L) =\left\{ \begin{pmatrix}-2r\\r\end{pmatrix}, r\in\mathbb{R}  \right\} \subset \mathbb{R}^2$$
Geometrically, $\ker(L)$ is the line travelling through the point $(-2,1)$ and the origin.
A: $L$ is an operator, which is defined as a mapping taking elements of one space into another (often the same space). So suppose $L:V\to W.$ Then from a set theory perspective, just like with functions, we can define $L$ by all the ordered pairs $(v, L(v))$ for all $v\in V.$ The null space is the set of all $v\in V$ that $L$ sends to $0\in W.$ That is, $\operatorname{null}(L)=\{v\in V:L(v)=0_W\}.$
