= Coefficient Banach space for normalized even maps
{title2=$p(x)=1-x^2h(x^2),\quad\|h\|=\sum_n|h_n|$}
Fix $R>1$ and write $h(z)=\sum_{n\geq0}h_n((z-1)/R)^n$ with absolutely summable real coefficients. The coefficient space is a <Banach space> under $\|h\|=\sum_n|h_n|$. The normalized even maps $p(x)=1-x^2h(x^2)$ form an affine copy of that space and are analytic where $|x^2-1|<R$. On any smaller coefficient disk, the tail beyond degree $N$ is bounded by $q^{N+1}\sum_{n>N}|h_n|$ for a radius ratio $q<1$. Analytic composition and <Cauchy estimates> on smaller domains permit certified derivative and truncation bounds. A small ball about an admissible map must additionally preserve the quadratic critical maximum and restrictive-interval conditions; the coefficient norm alone does not impose them.
// Target: banach-space.bigb
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