For $1\le p<\infty$, with $\mu$ $\sigma$-finite when $p=1$, continuous linear functionals on $L^p$ are integration against $L^q$ functions.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space, let \(1\leq p<\infty\), and let \(q\in(1,\infty]\) satisfy \(\frac1p+\frac1q=1\). If \(p=1\), assume that \(\mu\) is $\sigma$-finite. Then the map \[ L^q(X,\mu)\to \dualof{L^p(X,\mu)},\qquad g\mapsto\qty{f\mapsto\int_X fg\dmu}, \] is an isometric isomorphism onto the space \(\dualof{L^p(X,\mu)}\) of continuous linear functionals on \(L^p(X,\mu)\) with the dual norm [@Fol13].
State the Riesz representation theorem for $\dualof{L^p(X)}$.