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I'm faced with the following problem: Let $X$ and $U$ be Banach spaces, $M$ a bounded linear map from $X$ to $U$, and $M$ is onto. Prove: There exists some constant $C>0$, such that for all sequence $y_n$ in $U$ converging to $y_0$, we can find a sequence $x_n$ in $X$, such that $Mx_n=y_n$, and $x_n$ is convergent with $\|x_n\|\le C\|y_n\|$.

My idea is as follows: It is trivial if $M$ is injective, by bounded inverse theorem. If not, we can consider the quotient space $X/N_M$, here $N_M$ is the null space of $M$. Then we can find a sequence $x_n$ in X,such that $\operatorname{dist}(x_n,N_M)\le C\|y_n\|$, and $\operatorname{dist}(x_n-x_0,N_M)$ tends to zero for some $x_0$ in $X$. Now it remains to take a sequence in $N_M$ properly so that we can pass to $X$. But I have difficulty in doing this. Thanks for help!

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You should look at the proof of open mapping theorem en.wikipedia.org/wiki/… –  Norbert Jan 21 '12 at 9:40
    
@cheng: I've tried to edit your post - adding LaTeX markup, paragraphs; hopefully making it a little more readable. Please, check whether I did not unintentionally change the meaning in some place and edit the post again, if necessary. –  Martin Sleziak Jan 21 '12 at 10:06
    
Exactly what I mean.Thanks. –  cheng Jan 22 '12 at 5:23
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