= Hadamard remainder near a compact set
{c}
{title2=$\sup_{h\in K^\delta}|R_t(h)|<\epsilon\text{ for sufficiently small }|t|$}
For a <Hadamard differentiable> functional with continuous linear derivative $L$, define $R_t(h)=(\Phi(s+th)-\Phi(s))/t-L(h)$. For every compact $K$ and positive $\epsilon$, some positive $\delta,a$ make $|R_t(h)|<\epsilon$ whenever $\operatorname{dist}(h,K)<\delta$ and $0<|t|<a$. If this failed, choose $t_j\to0$, directions within $1/j$ of $K$ and remainders at least $\epsilon$. Compactness gives a convergent subsequence of nearby points of $K$, hence convergent directions. The derivative definition and continuity of $L$ give remainder tending to zero, a contradiction. With a tight weak limit, the <Portmanteau theorem> makes random directions lie near a suitable compact set with arbitrarily large probability. Thus this lemma proves the <functional delta method> without assuming separability of the entire normed space or uniform differentiability on its whole unit ball.
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