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Currently, I've started to take Analysis 2 in school and we are doing Euclidean Spaces. There is an example that I'm trying to prove but I can't wrap my mind around.

$x, y \in \mathbb R^d$

$\langle x,y \rangle:= \sum^d_{j=1}x_j-y_j$

So, there are 4 rules for this to be an inner product. Non-negativity, definite, symmetric, linearity. But I think this isn't non-negative; am I wrong? $y$ vector can always be bigger than $x$ and that can make the sum negative? Or am I missing something?

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There is a very easy way to see that this is not an inner product. Note that it often happens that inner products are negative for given choices of $x$ and $y$. However, what happens when you compute $\langle x,x\rangle$ for any $x$? Is this a positive number for all $x\neq 0$?

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  • $\begingroup$ Aha! I was trying to prove that it was an inner product that I didn't even think about that. Thank you so much! $\endgroup$
    – Xia
    Feb 9, 2019 at 11:43
  • $\begingroup$ @70pr4k You're welcome. $\endgroup$
    – Matt Samuel
    Feb 9, 2019 at 11:46
  • $\begingroup$ I'd say simple, not easy. Saying a problem is easy to solve is very relativistic. $\endgroup$
    – user273865
    Feb 12, 2019 at 1:09

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