every orthonormal set is linearly independent

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

Proof. We denote by $⟨\cdot ,\cdot ⟩$ 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.,$S=\{e_1,\mathrm{\dots },e_n\}$ for some $n$.Suppose

$${\lambda }_1{e}_1+\mathrm{\cdots }+{\lambda }_n{e}_n=0$$

for some scalars ${\lambda }_i$ (belonging to the field on theunderlying vector space of $L$). For a fixed $k$ in $1,\mathrm{\dots },n$,we then have

$$0=⟨e_k,0⟩=⟨e_k,{\lambda }_1{e}_1+\mathrm{\cdots }+{\lambda }_n{e}_n⟩={\lambda }_1⟨e_k,e_1⟩+\mathrm{\cdots }+{\lambda }_n⟨e_k,e_n⟩={\lambda }_k,$$

so ${\lambda }_k=0$, and $S$ is linearly independent.Next, suppose $S$ is infinite (countable or uncountable). To provethat $S$ is linearly independent, we need to show thatall finite subsets of $S$ are linearly independent. Since anysubset of an orthonormal set is also orthonormal, the infinite casefollows from the finite case. $\mathrm{\square }$