Talk:Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space

Sufficient Condition

Sufficient condition needs a proof. I found a proof in StackExchange question Is $Y$ is a Banach space if $B(X,Y)$ is a Banach space?. --Hbghlyj (talk) 16:21, 19 March 2024 (UTC)

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

\(\ds f(x_0)\)

\(=\)

\(\ds \lim_{n\to\infty}f_{y_n}(x_0)\)

\(\ds \)

\(=\)

\(\ds \lim_{n\to\infty}\varphi(x_0)y_n\)

\(\ds \)

\(=\)

\(\ds \lim_{n\to\infty}y_n\)