Note that for non-compact locally compact groups "G" the space formula_77 does not contain formula_104, so the Fourier transform of general formula_75-functions on "G" is "not" given by any kind of integration formula (or really any explicit formula). To define the formula_75 Fourier transform one has to resort to some technical trick such as starting on a dense subspace like the continuous functions with compact support and then extending the isometry by continuity to the whole space. This unitary extension of the Fourier transform is what we mean by the Fourier transform on the space of square integrable functions.