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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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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 $Y$. –  Martin Sleziak Jun 7 '12 at 11:24
    
And BTW, since you mentioned filters and nets, they can be used to desribe limit at a point; but I believe that is interesting only as an illustration that this is one of the things which is generalized by nets and filters, it is much easier to use the usual definition I linked in the above comment. –  Martin Sleziak Jun 7 '12 at 11:28
    
Yes, thank you, the definition you are pointing at is quite clear and makes sense. –  vkubicki Jun 7 '12 at 11:33

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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 $Y$.

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