Bishop phelps

Let $X$ be real, let $C$ be as in claim 1, and fix $f\in X^{\ast }$ and $\epsilon >0$. By [F1] there are $v\in C$ and $g\in X^{\ast }$ such that $\mathrm{\parallel }g-f\mathrm{\parallel }<\epsilon$ and $g(c)\le g(v)$ for every $c\in C$. Thus $g(v)={\sup }_{C}g$, and arbitrary $f$ and $\epsilon$ prove the asserted norm density.

Now let $X$ be complex and write $X_{\mathbb{R}}$ for its realification. For $u\in (X_{\mathbb{R}}{)}^{\ast }$ define $U_u(x)=u(x)-iu(ix)$. Real linearity gives $U_u(ix)=i{U}_u(x)$ and $\mathrm{Re}{U}_u=u$. For any $x$, choose a unit scalar $a$ with $a{U}_u(x)=\mathrm{\mid }{U}_u(x)\mathrm{\mid }$ when the value is nonzero, and take $a=1$ otherwise. Then $\mathrm{\mid }{U}_u(x)\mathrm{\mid }=u(ax)\le \mathrm{\parallel }u\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }$, while $\mathrm{\mid }u(x)\mathrm{\mid }\le \mathrm{\mid }{U}_u(x)\mathrm{\mid }$; therefore $U_u\in X^{\ast }$ and $\mathrm{\parallel }{U}_u\mathrm{\parallel }=\mathrm{\parallel }u\mathrm{\parallel }$. The correspondence is real-linear, and for complex-linear $f$, $U_{\mathrm{Re}f}=f$.

Take $C=B_X$ in step 1.1. Since $B_X$ is symmetric, ${\sup }_{x\in B_X}g(x)={\sup }_{x\in B_X}\mathrm{\mid }g(x)\mathrm{\mid }=\mathrm{\parallel }g\mathrm{\parallel }$ by [F2]. Hence the approximating $g$ satisfies $g(v)=\mathrm{\parallel }g\mathrm{\parallel }$ at some $v\in B_X$ and is norm-attaining. This includes $g=0$, which attains norm zero at zero.

Fix $f\in X^{\ast }$ and $\epsilon >0$. Apply the real claim 1 to the same set $B_X$ inside $X_{\mathbb{R}}$ and to $\mathrm{Re}f$. It gives a real functional $u$ and $v\in B_X$ with $\mathrm{\parallel }u-\mathrm{Re}f\mathrm{\parallel }<\epsilon$ and $u(x)\le u(v)$ on $B_X$. Put $G=U_u$. Step 1.2 applied to $u-\mathrm{Re}f$ gives $\mathrm{\parallel }G-f\mathrm{\parallel }<\epsilon$. Because the complex unit ball is symmetric, [F2] and step 1.2 give $u(v)={\sup }_{B_X}u=\mathrm{\parallel }u\mathrm{\parallel }=\mathrm{\parallel }G\mathrm{\parallel }\mathrm{.}$ But $\mathrm{Re}G(v)=u(v)=\mathrm{\parallel }G\mathrm{\parallel }$ and $\mathrm{\mid }G(v)\mathrm{\mid }\le \mathrm{\parallel }G\mathrm{\parallel }$, so $G(v)=\mathrm{\parallel }G\mathrm{\parallel }$ and $G$ attains its norm.

The three density assertions follow from steps 1.1, 2.1 and 2.2. If $X=\{0\}$, its unique functional is zero and already norm-attaining. The proof uses DC and HB only through [F1]; the realification and complexification are explicit and use no choice. The general convex-set conclusion remains real, while the complex conclusion is exactly the unit-ball norm-attainment assertion.