= Solution
For positive integer indices, distinct members are <orthogonal> and each has nonzero squared <norm> $\pi$. Thus $\{\sin(nx):n\ge1\}$ is an <orthogonal basis> in the <Hilbert space> of <odd> <L2 functions> on $(-\pi,\pi)$. The zero-index member must be omitted. More precisely this is a complete <Fourier sine basis>, with generally infinite <L2 norm convergence>, rather than a <Hamel basis> with finite linear combinations. A claim about arbitrary real <functions> requires suitable integrability and a specified sense of <convergence>.
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