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I was reading Reed and Simon Methods of Mathematical Physics Volume 1 and have a question about a small their proof of the open mapping theorem for Banach spaces.

Let $T:X\rightarrow Y$ be a bounded linear map between Banach spaces. The reduce the problem to checking the statement with open sets around the origin (which I am fine with). Then they say the following sentence which I cannot figure out:

"Since neighborhoods contain balls it is sufficient to show that $T[B_r^X]\supset B_{r'}^Y$ for $$B_r^X=\{x\in X:||x||<r\}\text{ "}$$

I guess I am confused with how does one open ball a subset $T[B_r^X]$ imply that all of $T[B_r^X]$ is open.

Thank you for any help. It is much appreciated.

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If $y \in T[B_r^X]$, then $y = Tx$ for some $x \in B_r^X$. Now $x + B_s^X \subset B_r^X$ where $s = r - \|x\|$, and $ y + T[B_s^X] =T[x + B_s^X] \subset T[B_r^X]$. So if $B_{s'}^Y \subset T[B_s^X]$, $y + B_{s'}^Y$ is an open neighbourhood of $y$ contained in $T[B_r^X]$.

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Thank you. This is much appreciated. –  user45150 Feb 12 '13 at 6:39

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