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Linear algebra studies vector spaces and linear mappings between those spaces.

What tools do we use for NON-linear mappings between vector spaces?

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    $\begingroup$ Usually one studies continuous mapping between topological spaces. Unless the space are vector spaces, linear-mappings between them are not defined. $\endgroup$ – Zuriel Dec 12 '14 at 10:49
  • $\begingroup$ What would a linear map between topological spaces be? $\endgroup$ – Tobias Kildetoft Dec 12 '14 at 10:49
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    $\begingroup$ The classification of maps as linear and non-linear is similar to the classification of fruit as bananas and non-bananas :-) $\endgroup$ – Georges Elencwajg Dec 12 '14 at 10:50
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    $\begingroup$ @GeorgesElencwajg Hmm, so vector spaces are to topological spaces as monkeys are to...? $\endgroup$ – Tobias Kildetoft Dec 12 '14 at 10:56
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    $\begingroup$ @Zuriel and Tobias Kildetoft: you are correct. My second question is nonsense, I will delete it. $\endgroup$ – SiXUlm Dec 12 '14 at 11:11
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The field you are concerned with is called nonlinear functional analysis. (Calculus of variations can be considered to fall in here as well.) There are many tools in nonlinear functional analysis, though they are often fairly specialized since there are so many nonlinear mappings between topological vector spaces.

Common tools involve fixed-point theorems/root-finding theorems (think Newton's method, Brouwer fixed point theorem and its generalizations), generalizations of differentiability, bifurcation theory, and Morse theory/analysis of critical points, and this is very far from an exhaustive list. The tools by nature have to be fairly topological or analytic, because you can no longer take advantage of the algebraic properties a linear mapping enjoys.

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