Theorem: An orthonormal set of vectors in an inner product space is linearly independent.

Proof. We denote by $\langle \cdot, \cdot \rangle$ the inner product of $L$ . Let $S$ be an orthonormal set of vectors. Let us first consider the case when $S$ is finite, i.e., $S=\{e_1,\ldots, e_n\}$ for some $n$ . Suppose $$ \lambda_1 e_1 + \cdots + \lambda_n e_n =0$$ for some scalars $\lambda_i$ (belonging to the field on the underlying vector space of $L$ ). For a fixed $k$ in $1,\ldots, n$ , we then have $$0=\langle e_k,0\rangle = \langle e_k,\lambda_1 e_1 + \cdots + \lambda_n e_n \rangle = \lambda_1 \langle e_k,e_1\rangle + \cdots + \lambda_n \langle e_k,e_n\rangle = \lambda_k,$$

so $\lambda_k=0$ , and $S$ is linearly independent. Next, suppose $S$ is infinite (countable or uncountable). To prove that $S$ is linearly independent, we need to show that all finite subsets of $S$ are linearly independent. Since any subset of an orthonormal set is also orthonormal, the infinite case follows from the finite case.