## Question on isometries

Dear Professor,

Firstly thanks for taking the time out to reply to our messages, we really appreciate the help and guidance from yourself. I wanted to ask the following short question about the 2201 notes:

On page 56 Thm 5.4.150: after showing that and , can’t we simply show by saying:

and proceeding exactly as the last line of the proof at the bottom?

Thanks in advance.

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Reply:

Not quite. Although with a bit more argument it might be OK. In `the last line of the proof at the bottom,’ note that an unwritten step in the argument says

for *all* , and then one sets . This wouldn’t be admissible if we just had . Try it out to see clearly for yourself what I mean.

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