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!
2026-03-29 18:31:04.1774809064
orthonormal basis in $L^2$ space
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