Is $Y$ is a Banach space if $B(X,Y)$ is a Banach space?

Question

Let $X$ and $Y$ be normed linear spaces, $X\ne\{0\}$. Let $B(X,Y)$ be the collection of all bounded linear operators from $X$ into $Y$ with the operator norm. Suppose that $B(X,Y)$ is a Banach space. Show that $Y$ is a Banach space.

Answer 1

Pick a Cauchy sequence in $Y$, say $(y_n)$. Define given $y\in Y$ the mapping $f_y:X\to Y$ that sends $x\to \varphi(x)y$ where $\varphi\in X^\ast$ is a fixed nonzero functional, let $x_0$ be such that $\varphi(x_0)=1$. Note that $\lVert f_y\rVert \leqslant \lVert \varphi\rVert \lVert y\rVert $ so the $f_y $ are bounded, of course they are linear. In fact the equality holds. It follows that the $f_{y_n}$ are Cauchy in $B(X,Y)$ if the $(y_n)$ are Cauchy in $Y$. Let $f$ be such that $f_{y_n}\to f$. Then $$f(x_0)=\lim_{n\to\infty}f_{y_n}(x_0)=\lim_{n\to\infty}\varphi(x_0)y_n=\lim_{n\to\infty}y_n$$