# Continuous linear operator with noncontinuous inverse [duplicate]

This question already has an answer here:

This question gave an example of a continuous $f: E \rightarrow F$ which is bijective but has noncontinuous inverse. In the example, neither $E$ nor $F$ was a Banach space, are there any examples where $E$ or $F$ is a Banach space (of course, none exist if both $E$ and $F$ are Banach, by an application of the open mapping theorem).

## marked as duplicate by Nate Eldredge, TMM, Chris Janjigian, user85798, user61527 Mar 13 '14 at 3:05

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• See this. – David Mitra Mar 12 '14 at 23:37

## 1 Answer

Let $E=C([0,1],\Bbb R)$ (continuous maps frome $[0,1]$ to $\Bbb R$) with $\|.\|_{\infty}$ and $\|.\|_1$ and identical mape : $f \mapsto f$ frome $(E,\|.\|_{\infty})$ to $(E,\|.\|_{1})$

• But, are both Banach spaces? – Berci Mar 12 '14 at 23:47
• i am editing with an other example. the question says E$or$F$Banach ... – Mohamed Mar 13 '14 at 0:14 • It's well known that$(E, \|\cdot\|_\infty)$is Banach and$(E, \|\cdot\|_1)\$ is not. – Nate Eldredge Mar 13 '14 at 0:32