= Weak holomorphy implies norm holomorphy
For a <weakly holomorphic Banach-valued function>, choose a closed disk contained in its domain. Each continuous functional is bounded on the image of that disk by scalar continuity. The <Uniform boundedness principle>, applied to its evaluation functionals in the bidual, gives a common bound $M$ on the <norms> of the image vectors. Cauchy's formula on the disk's boundary then bounds the scalar difference between difference quotients at increments $h,k$ by a constant times $M\|\varphi\||h-k|$, uniformly for sufficiently small increments. Explicitly the difference kernel is
$$
\frac{h-k}{(\zeta-z)(\zeta-z-h)(\zeta-z-k)}.
$$
The <Hahn-Banach theorem> dual <norm> formula converts this into the same <norm> bound for the vector difference quotients. They are Cauchy, so completeness gives a <norm> derivative. This also explains why weak holomorphy suffices without a separately assumed <norm>-continuity hypothesis.
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