For a $\sigma$-finite measure space $(\Omega,\mathscr{F},\mu)$, is $L^2\subset L^1$ always true?
2026-04-18 08:12:39.1776499959
$L^2$ and $L^1$ space problem
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No. $L^2(\mathbb R^+) \not\subset L^1(\mathbb R^+)$. You need a finite measure space. An example for my claim is $$f(x) = \frac1{x+1}$$ $f\in L^2(\mathbb R^+) \setminus L^1(\mathbb R^+)$.