Complex Lp duality from real Lp duality

Regard real $L^p(\mu )$ as the real-valued subspace of complex $L^p(\mu ;\mathbb{C})$. The maps $A(u)=\mathrm{Re}\mathrm{\Lambda }(u)$ and $B(u)=\mathrm{Im}\mathrm{\Lambda }(u)$ are bounded real-linear functionals there, with $\mathrm{\mid }A(u)\mathrm{\mid },\mathrm{\mid }B(u)\mathrm{\mid }\le \parallel \mathrm{\Lambda }\parallel \parallel u{\parallel }_p$. Applying [F1] twice gives real $a,b\in L^q(\mu )$ such that $A(u)=\int ua$ and $B(u)=\int ub$ for every real $u\in L^p$. Put $h=a+ib\in L^q(\mu ;\mathbb{C})$; component inequalities in [F3] make its $q$-norm finite.

For real-valued $u$, componentwise complex integration gives $\mathrm{\Lambda }(u)=A(u)+iB(u)=\int u(a+ib)=\int uh$. If $f=u+iv$ is an arbitrary complex $L^p$ class, [F3] puts its real and imaginary parts in real $L^p$, and complex linearity gives $\mathrm{\Lambda }(f)=\mathrm{\Lambda }(u)+i\mathrm{\Lambda }(v)=\int uh+i\int vh=\int fh$. All identities depend only on a.e. classes by the quotient and integration conventions in [F2]–[F3].

Hölder [F3] gives $\mathrm{\mid }\int fh\mathrm{\mid }\le \parallel f{\parallel }_p\parallel h{\parallel }_q$, hence $\parallel \mathrm{\Lambda }\parallel \le \parallel h{\parallel }_q$. If $h=0$ a.e., step 2.1 gives $\mathrm{\Lambda }=0$ and equality follows. Otherwise define $v=0$ on $\{h=0\}$ and $v=\mathrm{\mid }h{\mathrm{\mid }}^{q-2}\bar{h}$ where $h\mathrm{\ne }0$. Then $\mathrm{\mid }v\mathrm{\mid }=\mathrm{\mid }h{\mathrm{\mid }}^{q-1}$ and $vh=\mathrm{\mid }h{\mathrm{\mid }}^q$ pointwise. Since $(q-1)p=q$, [F2]–[F3] give $v\in L^p$, $\parallel v{\parallel }_p=\parallel h{\parallel }_q^{q-1}$, and $\mathrm{\Lambda }(v)=\int \mathrm{\mid }h{\mathrm{\mid }}^q=\parallel h{\parallel }_q^q$. Testing on $v\mathrm{/}\parallel v{\parallel }_p$ proves $\parallel \mathrm{\Lambda }\parallel \ge \parallel h{\parallel }_q$, including the closed unit-norm endpoint.

If $k\in L^q(\mu ;\mathbb{C})$ gives the same pairing functional, put $d=h-k$. Then $\int fd=0$ for every $f\in L^p$. If $d$ were nonzero, the phase test of step 3.1 with $d$ in place of $h$ would produce $v\in L^p$ with $\int vd=\parallel d{\parallel }_q^q>0$, a contradiction. Thus $d=0$ in $L^q$, so the density is unique.

Steps 2.1–4.1 prove existence, equality of norms, and uniqueness. Countable Choice is used only inside the real arbitrary-measure representation [F1], whose construction makes countably many local choices; applying that theorem to $A$ and $B$ requires only two instances and no stronger choice principle. The empty and zero-measure spaces have only zero $L^p$ classes and are covered by the $h=0$ branch of step 3.1; the forbidden endpoints $p=1,\mathrm{\infty }$ never enter because $1<p,q<\mathrm{\infty }$.