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10:41 AM

Can someone get me a dupe for this? Approach0 not quite giving it to me, I'll try to search a little more.

Let $V$ be an inner product space over $\mathbb{R}$, and let $T : V \to V$ be a linear map. I'd like to prove the following statement: If $||T(v)|| = ||v||$ for all $v \in V$, then $\langle u,v \rangle = \langle T(u), T(v)\rangle $ for all $u,v \in V$. I was advised to consider $||T(u + v)||$ but...

By searching for linear preserves norm among the posts tagged inner-products I found some related posts:

For example: 1626701 $T$ preserves inner product iff $\|T\alpha \|=\|\alpha\|$, 3696581: For any two vectors $v_1$ and $v_2$, $\langle v_1, v_2 \rangle = \langle T (v_1), T (v_2) \rangle$ if $T$ is norm-preserving.

I suppose that there are also some other questions on the same topic.

Here you go : $T$ preserves inner product iff $\|T\alpha \|=\|\alpha\|$ — Teresa Lisbon 11 mins ago

in CURED, 27 secs ago, by Martin Sleziak

@TeresaLisbon I have mentioned some in the searching chatroom - although I see from your comment that you have already found something.

Martin Sleziak

11:32 AM

in CURED, 2 mins ago, by Teresa Lisbon

@MartinSleziak Thank you : I found the one I needed. I didn't find it from Approach0, but rather went with a decent search text on Google : norm preserving implies inner product preserving. I could have chosen any of the posts that came in that list, thanks for your efforts. I'll also use the chatroom when I can.

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