Consider $L_p$-function spaces, $p\ge 1$. Let $X$ be the bidual of $L_1$. What is the bidual of $L_1+L_p$? Is it $X+L_p$? It is quite weird since I don't know whether $X$ is a function space or not. However, I know that $L_1^{**}=L_1\oplus_1 V_0$ for some $V_0 \subset L_1^{**}$ ($\oplus_1$ indicates that the norm is the sum of the norms on $L_1$ and $V_0$, see Yosida, K., Hewitt, E., Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66 (1952) or Takesaki, M. Theory of Operator Algebras. Chapter III.2.14). So, is it true that $L_1^{**}= (L_1+L_p)\oplus_1 V_0$?
2026-03-24 22:15:45.1774390545
What is the bidual of $L_1+L_p$?
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