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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!

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    Well I stared at the statement a bit but it seems kind of counterintuitive. You think I should try to prove it for realsies?2011-11-28

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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.

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    Well I guess when you put it that way, ||Lx_2 - Lx_1|| < ||L|| ||x_2 - x_1|| \to 0 for bounded $L$. Thanks! I think I was just staring at the wrong condition.2011-11-28