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This is an exercise from Royden's Real Analysis. Suppose $X$ is a Banach space, there is a continuous, linear, open map from $X$ onto a normed linear space $Y$. Show that $Y$ is Banach.

GEdgar
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So Wai Chung
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1 Answers1

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Hints. Let $T$ be your linear map. Show first that there exists a finite constant $C$ such that, for any $y\in Y$, one can find $x\in X$ such that $Tx=y$ and $\Vert x\Vert\leq C\,\Vert y\Vert$. Use this to show that any series $\sum y_n$ in $Y$ such that $\sum_0^\infty \Vert y_n\Vert<\infty$ is convergent in $Y$.

Etienne
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  • Etienne, thanks for help. I was thinking constructing an homeomorphism between X/KerT and Y which I thought was possible. – So Wai Chung Mar 12 '14 at 20:57
  • This is indeed possible. You can use open-ness to show that the induced map from $X/\ker(T)$ onto $Y$ has a continuous inverse. – Etienne Mar 12 '14 at 21:04