Are bounded linear maps continuous? 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:


*

*$\Lambda$ is bounded.

*$\Lambda$ is continuous.

*$\Lambda$ is continuous at one point of $X$.


The proof is very straightforward.
