If a set S is linearly independent with each element having norm 1, is S orthonormal?

Question

Let $V$ be an inner product space.

If a set $S \subset V $ is linearly independent and each member of $S$ has norm 1, is $S$ orthonormal?

I know that if you remove the norm condition, it is not true that $S$ is an orthogonal set.

Answer 1

No. In $\Bbb R^2$ with the usual Euclidean metric, for example, consider $$ S = \bigg\{ (0,1), \bigg( \frac35,\frac45 \bigg) \bigg\}. $$