$\begingroup$

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!

$\endgroup$ 3

1 Answer

$\begingroup$

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.

$\endgroup$ 3

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy