A real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. The set of all measurable functions that are square-integrable forms a Hilbert space.
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... the chair of philosophy at Halle for two years, he became, in 1833, professor at the university of Königsberg[?], where he remained till his death. In his last years he ...