Details

Autor(en) / Beteiligte

• Boris Buffoni
• John Toland

Titel

Calculus in Banach Spaces

Ist Teil von

• Analytic Theory of Global Bifurcation, 2016, p.21

Ort / Verlag

Princeton University Press

Erscheinungsjahr

2016

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We turn to our main topic, nonlinear operators between Banach spaces. “Big O” and “little o” notation. Suppose thatfandgare functions defined from a neighbourhood ofain a Banach spaceXto a Banach spaceY. We write\[f(x)=g(x)+o(x-a)\text{ as }x\to a\text{ if }\underset{x\to a}{\mathop{\lim }}\,\frac{\left\| f(x)-g(x) \right\|}{\left\| x-a \right\|}=0,\]and\[f(x)=g(x)+O(x-a)\text{ as }x\to a\text{ if }\underset{x\to a}{\mathop{\lim \text{sup}}}\,\frac{\left\| f(x)-g(x) \right\|}{\left\| x-a \right\|}<\infty .\] Suppose thatXandYare Banach spaces,U⊂Xis open, andF:U→Y. Definition 3.1.1The function F is Fréchet differentiable at x 0∈U if there exists A∈ L(X,Y)such that \[\underset{0<\left\| h \right\|\to 0}{\mathop{\lim }}\,\frac{\left\| F({{x}_{0}}+h)-F({{x}_{0}})-Ah \right\|}{\left\| h \right\|}=0.\] If such an operator A exists, it is unique and is called the Fréchet