Note that the converse is not true in . That is,
does not imply
in . For a counterexample, consider ,
, in which case

while
.

For real vectors
, the Pythagorean theorem Eq. (5.1)
holds if and only if the vectors are orthogonal. To see this, note
that, from Eq. (5.2), when the Pythagorean theorem holds, either
or is zero, or
is zero or purely imaginary,
by property 1 of norms (see §5.8.2). If the inner product
cannot be imaginary, it must be zero.

Note that we also have an alternate version of the Pythagorean
theorem: