# Show that the following is an inner product

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?

## 1 Answer

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 $$x\neq 0$$?

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