Kernel and Range of a linear operator

Hey guys,

Let $\displaystyle A$ be a linear operator from a vector space $\displaystyle X$ into a vector space $\displaystyle Y$, and define also $\displaystyle \mathcal{B}(X,Y)$ as the vector space of all bounded linear operators. Now, a set of notes claims the following

"If $\displaystyle X$ and $\displaystyle Y$ are normed spaces, and $\displaystyle A \in \mathcal{B}(X,Y)$, then Ker$\displaystyle (A)$ is closed; this is not necessarily true for Ran$\displaystyle (A)$."

Why is this the case?

Thanks a lot in advance,

HTale.