Abstract
It is shown that by starting with a general form of the Peano kernel theorem which makes no reference to the interchange of linear functionals and integrals, the most general results can be obtained in an elementary manner. In particular, we classify how the Peano kernels become increasingly smooth and satisfy boundary (or equivalently moment) conditions as the linear functionals they represent become continuous on wider classes of functions. These results are then used to give new representations of the continuous duals of .
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