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-2
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How can one prove that $S( \mathbb{R}^{d}) $, the Schwartz space, is dense in $L^{2}(\mathbb{R}^{d}) $? A slightly stronger result is that $C_{0}^{\infty}( \mathbb{R}^{d}) $, the set of infinitely differentiable functions with compact support, is dense in $L^{2}(\mathbb{R}^{d}) $, but I think this is probably easy to show once it has been proved that $S( \mathbb{R}^{d}) $ is dense in $L^{2}(\mathbb{R}^{d}) $. |
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2
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This is an easy homework problem and not appropriate for MO. |
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0
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Typically, one proves first that simple functions are dense, then extends this characterization to step functions, then smooth functions with compact support (this involves the classic "bump" construction). |
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