Let $H$ a linear space with inner product.
An orthonormal system $\{e_1, e_2, \dots \}$ is called complete in $H$ if $x=0$ is the only element that satisfies the relations: $$(x, e_n)=0, \ \ \ n=1,2, \dots$$
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Isn't it $ \ \ \ (e_i,e_j)=0 , \ \ \ \forall i \neq j \ \ $ if we have an orthonormal basis??
Why is $x=0$ the only elements that satisfies the above relations??
$$(x, e_n)=0, \ \ \forall n\geq 0$$ while $e_j$ does not satisfy this because $$(e_j, e_j)=1$$ i.e., if we set $x=e_j$ and $n=j$ then $$(x, e_n)\neq 0$$