Examples of incomplete normed spaces of continuous linear maps between two normed spaces [duplicate]

Sketch (fill in the details):

Suppose $(T_n)$ is Cauchy in $B(X,Y)$. Then $(T(x)_n)$ is Cauchy in $Y$ for every $x\in X\Rightarrow \lim_{n \to \infty} T_n(x) = y_x.$ This defines a map $x\mapsto y_x$, which is evidently linear. Furthermore, $\|T_m(x)-T_n(x)\|\le \|T_m-T_n\|\|x\|\le \epsilon \|x\|$ if $m,n$ are large enough. Let $m\to \infty$. Then, $\|T-T_n\|\le \epsilon $, which shows that $T_n\to T$ and that since $T-T_n\in B(X,Y)$ so $T=T_n+T-T_n$ is, too.

Now, suppose $(y_n)$ Cauchy. First choose a $\phi\in X^*$ and an $x_0\in X$ such that $\phi(x_0)=1.$ For each $y\in Y$, define a map $g_y:X\to Y:x\mapsto \phi(x)y$ and show that $(g_{y_n})\subseteq B(X,Y)$ is Cauchy and therefore $g_{y_n}\to g\in B(X,Y)$. Finally, show that $y_n\to g(x_0).$