In real inner product space, I'm trying to show the following proof:
$<u+v, u-v>\\ = <u,u>+<u,-v>+<v,u>+<v,-v>\\ = ||u||^2 - <u,v> + \overline{<u,v>}-||v||^2\\ =||u||^2 - <u,v> + <u,v>-||v||^2\\ = ||u||^2 -||v||^2$
Is it necessary to do it from right to left again? Or is it sufficient if I switch = to <=> and say each step is an if-and-only-if?
Thanks!
In mathematics you never switch "$=$" and "$\iff$", since they do not apply in the same contexts. Equality ("$=$") holds between expressions designating values, which can be of any type, but not truth values. Equality is an equivalence relation, so if you have shown $x=y=z$ then you have also shown for instance $z=x$. On the other hand logical equivalence ("$\iff$") holds between statements, which are expressions with a truth value (as for instance "$x\leq 3$") and it expresses that the statements have the same truth value. There are actually two uses of this: one where you claim that all statements so linked by logical equivalence are true, usually the initial one because it is evident or holds by assumption, so that the truth of the others follows from the logical equivalence, and another use where each statement may be true or false (because there are free variables whose values are not known) and you are only asserting that they express the same logical condition; which one applies is usually clear from the context. When using $\iff$ you must at each step verify logical equivalence (see that one statement implies the other and vice versa), and then you may use the reasoning in either direction. If you are only affirming implication in one direction you must use $\implies$ instead, and this does not give you logical equivalence.
In your case there are no statements to consider (other than the implicit one you are making that the equalities hold true), so there is no place for "$\iff$".