Let $X,Y$ be normed spaces, $B(X,Y)$ the normed space of continuous linear maps between $X$ and $Y$. Are there normed spaces $X,Y$ such that $B(X,Y)$ is not complete?
It is well-known that $B(X,Y)$ is complete if $Y$ is complete, so the desired $Y$ should be incomplete.
Here is an attempt:
Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$,
\begin{align}
\mathbb{F^N}\mathrel{\mathop:}=\left\{\{x_k\}_{k\in\mathbb{N}}:x_k\in\mathbb{F} \mbox{ for all }k\in\mathbb{N}\right\},
\end{align}
\begin{align}
c_e(\mathbb{F})\mathrel{\mathop:}=\{\{x_k\}_{k\in\mathbb{N}}\in \mathbb{F^N}:\mbox{there exists } N\in\mathbb{N}\mbox{ with } x_k=0\mbox{ for all }k\geq N\},
\end{align}
then $c_e(\mathbb{F})$ endowed with the supremum norm
\begin{align}
\|\{x_k\}_{k\in\mathbb{N}}\|\mathrel{\mathop:}=\sup_{k\in\mathbb{N}}\lvert x_k\rvert
\end{align}
is an incomplete normed space ( I learn this from Christian Clason's
Introduction
to Functional
Analysis).
Is $B(c_e(\mathbb{F}),c_e(\mathbb{F}))$ complete or not?
