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in the Wikipedia article on Gâteaux derivative , the limit of a function between two topological vector spaces is taken. How is the limit defined on a topological space for a function ? I find articles on net and filters for corresponding notions on topological spaces, but those are limits for discrete sequences, right ?

Thank you,

JD

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    Yes, thank you, the definition you are pointing at is quite clear and makes sense.2012-06-07

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For the sake of having an answer I am posting my above comment as an answer - since the OP seems to be satisfied with this. (Judging by his comment.)

Limit of a function between two topological spaces $X$ and $Y$ at a point $p\in X$ is defined quite similarly as in the case of real functions, see Wikipedia. In the definition of Gateaux derivative, you use a function from $\mathbb R$ to $Y$.