Let $B(X, Y)$ be the set of bounded linear maps from $X$ to $Y$ (i.e. such that $\sup_{||x|| \leq 1} L(x) < \infty$). Is $L \in B(X, Y)$ continuous? What about if $X$ is a Banach space? What about if $Y$ is a Banach space?
Thank you!
Let $B(X, Y)$ be the set of bounded linear maps from $X$ to $Y$ (i.e. such that $\sup_{||x|| \leq 1} L(x) < \infty$). Is $L \in B(X, Y)$ continuous? What about if $X$ is a Banach space? What about if $Y$ is a Banach space?
Thank you!
Theorem 5.4 from Rudin's Real and Complex Analysis: For a linear transformation $\Lambda$ of a normed linear space $X$ into a normed linear space $Y$, the following are equivalent:
The proof is very straightforward.