Non-completeness of the space of bounded linear operators

Question

If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. $B(X,Y)$ is not complete?

Answer 1

Let $X = \mathbb{R}$ with the Euclidean norm and let $Y$ be a normed space which is not complete. You should find that $B(X, Y) \simeq Y$.