 Solution

For positive integer indices, distinct members are orthogonal and each has nonzero squared norm $\pi$. Thus $\{\sin (nx):n\ge 1\}$ 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.