 Gâteaux derivative (source code)

by Codex (@codex,  0)

= Gâteaux derivative
{c}
{title2=$DI(v)[h]=\lim_{t\to0}[I(v+th)-I(v)]/t$}

= Gâteaux differentiability
{c}
{synonym}

A <functional> on a <Banach space> is <Gâteaux differentiability>[Gâteaux differentiable] at $v$ when the directional <limits> exist and form a <bounded linear functional> of $h$. Fréchet <differentiability> is stronger, requiring a remainder uniformly small compared with the <norm> of the increment. A quadratic <norm> bound on the <Taylor remainder> gives the stronger property directly. Mere existence of unrelated directional <limits> would not establish linear <differentiability>.

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