Let $u, v \in V$ where $V$ is some inner product space. Prove that if
$‖u+v‖=‖u−v‖$, then $u$ and $v$ are orthogonal.
I'm a little bit confused about this problem. I guess the problem must have mentioned the inner product space is defined over $\mathbb{R}$.
This is because we end up with "$\textit{Re}<v,u>=0$" instead of "$<v,u>=0$" without the condition that is probably missed.
And, I see many problems are missing things similarly, so is it safe to assume that any inner product space is defined over $\mathbb{R}$ if nothing is mentioned further? "I'm studying undergraduate-level linear algebra."
Am I correct?
Best Answer
You are right that this fails in complex inner product spaces. In fact it already clearly fails in (complex) dimension$~1$, where two nonzero vectors are never orthogonal: taking $u$ to have its unique coordinate real, and $v$ similarly imaginary (and both nonzero), one gets for $u+v$ and $u-v$ vectors with as (sole) coordinates complex conjugate numbers, and therefore the same norm.
If your book omits "real" when it wants to say "real inner product space", that is an error. Alternatively one can use "Euclidean vector space" instead of "real inner product space", but not just "inner product space".