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Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space. Since $L^2(\Omega,\mathcal{F},\mathbb{P})$ is a subspace of $L^1(\Omega,\mathcal{F},\mathbb{P})$, is there a well-defined/studied projection operator from $L^1(\Omega,\mathcal{F},\mathbb{P})$ onto $L^2(\Omega,\mathcal{F},\mathbb{P})$?

If so, could someone provide a reference/ let me know why this topic is not commonly studied?

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  • $\begingroup$ Certainly not a continuous one, because $L^2(\Omega,\mathcal F,\Bbb P)$ (as a subspace) is dense in $L^1(\Omega,\mathcal F,\Bbb P)$. $\endgroup$ – Saucy O'Path Apr 1 at 10:50
  • $\begingroup$ Indeed, but is there some studied notion? $\endgroup$ – AIM_BLB Apr 1 at 11:00

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