Space of linear operators

Question

Let $F$ is non-Banach space. Then I need do prove that $L(E, F)$ is also non-Banach space.

Answer 1

If $E=\{0\}$ then the conclusion is false, so assume that $E\neq\{0\}$.

Let $(f_n)$ be a Cauchy sequence in $F$ which is not convergent. There exist a non -zero continuous linear functional $F$ on $E$. Define $T_n \in L(E,F)$ by $T_nx=F(x) f_n$. Note that $\|T_n-T_m\| \leq \|F\| \|f_n-f_m\| \to 0$. Hence $(T_n)$ is Cauchy in $L(E,F)$. If $\|T_n-T\| \to 0$ for some continuous linear map $T$ then $\|F(x) f_n-Tx\| \to 0$ for each $x$. Choosing $x$ such that $F(x)\neq 0$ we get the contradiction that $(f_n)$ is convergent.