Differentiation in Fréchet spaces

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in particular in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and <a href="/facts/Nonlinear_analysis/dVxYYVho">nonlinear analysis</a>, it is possible to define the <a href="/facts/Derivative_(generalizations)/mgWY23Sd">derivative</a> of a function between two <a href="/facts/Fr%C3%A9chet_space/9gXkQu2u">Fréchet spaces</a>.  This notion of differentiation, as it is <a href="/facts/Gateaux_derivative/dqdtTtpf">Gateaux derivative</a> between Fréchet spaces, is significantly weaker than the <a href="/facts/Fr%C3%A9chet_derivative/kyAlz3up">derivative in a Banach space</a>, even between general <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>.  Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from <a href="/facts/Calculus/MEmUTYoz">calculus</a> hold.  In particular, the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a> is true.  With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the <a href="/facts/Inverse_function_theorem/pIwcyc6X">inverse function theorem</a> called the <a href="/facts/Nash%E2%80%93Moser_inverse_function_theorem/fswoago4">Nash–Moser inverse function theorem</a>, having wide applications in nonlinear analysis and <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>.
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Differentiation in Fréchet spaces open-in-new

Differentiation in Fréchet spaces