Let $f \in \mathcal{S}(\mathbb{R})$, Schwartz class. Then is it true that, $$\|f\|_2\leq \|f\|_1$$?
2026-03-25 23:16:16.1774480576
Norm inequality in Schwartz class.
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Cannot be true. If this is true then the inequality holds for all continuous functions with compact support and then for all $f \in L^{1}$. Hence $L^{1}$ would be contained in $L^{2}$ which is false.