= Solution
Regard the spaces as real and set $u=\operatorname{Re}\phi$. Its real <norm> equals $\|\phi\|$, since rotating any test vector makes its functional value real and nonnegative without changing its <norm>. By the allowed real <Hahn-Banach theorem> extend $u$ to a real-linear $v:X\to\mathbb R$ with that <norm>, and define $\psi(x)=v(x)-iv(ix)$. Real linearity gives $\psi(ix)=i\psi(x)$, so $\psi$ is complex linear. On $Y$ this formula recovers $\phi$. Also $\operatorname{Re}\psi=v$; rotating a test vector again gives $|\psi(x)|\leq\|v\|\|x\|$. The reverse <norm> inequality follows by restriction to $Y$, proving \b[a <norm>-preserving complex extension].
We use two standard consequences: a point outside a <closed subspace> can be separated by a bounded functional annihilating that subspace, and the <norm> of a vector is the supremum of its pairings with dual unit-ball functionals. If $\psi\in X^*$ annihilates $Y$, the scalar <holomorphic function> $\psi\circ f$ vanishes at the prescribed sequence tending to zero. Its <identity theorem> makes it identically zero. Separation then gives \b[$f(z)\in Y$ for every $z\in D$].
For $\phi\in Y^*$ extend to $X^*$ with the same <norm> and put $F=\phi\circ f$. Then $F$ is holomorphic with $|F|\leq\|\phi\|$. For the understood nonzero increments $h,k$, the <Cauchy integral formula> on $|\zeta|=r$ gives
$$
\frac{F(z+h)-F(z)}h-\frac{F(z+k)-F(z)}k=\frac{h-k}{2\pi i}\int_{|\zeta|=r}\frac{F(\zeta)}{(\zeta-z)(\zeta-z-h)(\zeta-z-k)}d\zeta.
$$
Taking absolute values, bounding the contour length by $2\pi r$, and each denominator below by $r-|z|$, $r-|z|-|h|$, $r-|z|-|k|$ proves \b[exactly the stated estimate].
Fix $z,r$ and restrict $|h|,|k|<(r-|z|)/2$. Taking the supremum over $\|\phi\|\leq1$ bounds the <norm> of the difference of the two $Y$-valued difference quotients by a fixed constant times $|h-k|$. They are therefore Cauchy as $h\to0$. Since $Y$ is a <closed subspace> of a <Banach space>, it is complete. Consequently \b[the derivative exists as a <norm> limit in $Y$]. The labelled (a)-(c) are the simultaneous hypotheses, not separate claims to prove.
Back to article page
