• differenziale di Gâteaux
• spazi di Banach
• funzione implicita
• principio variazionale di Ekeland
• differenziale di Fréchet

Generalization to the Implicit Function Theorem to infinite dimensional spaces has proven to provide important results in a very wide area of applications (just to mention one, the Nash embedding theorem). The most recent result in this sense is due to I. Ekeland, who obtains a nice inverse function theorem for Fréchet spaces based on his Variational Principle. This result can be restated as a Implicit Function Theorem, and it still provides interesting improvements in the Banach case. A comparison with other similar results is made, starting from the "classical" Hildebrandt-Graves theorem, a theorem from Deimling, and the result of A. Tarsia, which makes use of Campanato "near operators" theory. Finally, some result with minimal hypothesis are found for local injectivity in the finite-dimensional case.