Smith's Theorem

Smith's Theorem states that in a context called a symmetric function space—where functions are measured based on size and rearrangements—any linear, continuous, and symmetric operator acting on this space can be approximated arbitrarily closely by a special kind of operator called a multiplication operator. Essentially, this means complex transformations that respect the space’s symmetry and linearity are, in practice, very similar to just multiplying functions by a fixed, measurable function, simplifying the understanding and analysis of such operators.