Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks in advance for your help!
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These questions may help you: http://math.stackexchange.com/questions/433635/from-lang-sl-2-orthonormal-bases-for-l2-x-times-y?rq=1 http://math.stackexchange.com/questions/105451/orthonormal-basis-for-product-l2-space – s.harp Mar 14 '16 at 12:57