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Let $p,q>1 :1/p+1/q=1$ and define the map $\phi_{g}: L^{p}(X, \mu) \rightarrow \mathbb{F}$ by $$ \phi_{g}(f):=\int_{X} f g d \mu $$ where $g\in L^q(X,\mu)$ (with $L^p$ and $L^q$ function spaces). I already know that the function defined above is linear, but I would like to know if I can argue that this is true without a formal proof?

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  • $\begingroup$ Multiplication is bilinear and the integral is linear. Hence $\phi_g$ must be linear. Is this what you are looking for? $\endgroup$
    – Klaus
    Aug 5 at 17:06
  • $\begingroup$ So effectively I have $\phi_{1}: L^{p}(X, \mu) \rightarrow L^{1}(X, \mu)$, defined by $\phi_{1}(f):=f g$, and $\phi_{2}: L^{1} \rightarrow \mathbb{F}$, defined by $\phi(f g):=\int_{X} f g d \mu$. Because $\phi_1$ and $\phi_2$ are both linear, the composition $\phi_{g}(f)=\left(\phi_{2} \circ \phi_{1}\right)(f)$ is linear? $\endgroup$
    – Logi
    Aug 5 at 17:20
  • $\begingroup$ $\phi_2(h) := \int_X h d\mu$ to be precise, but yes. $\endgroup$
    – Klaus
    Aug 5 at 17:25
  • $\begingroup$ Great, thank you. $\endgroup$
    – Logi
    Aug 5 at 17:37

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